USCG Captain's License Exam

Advanced Ship Stability & Loading

Q: What is the metacentric height (GM) and how is it calculated?

Metacentric height (GM) is the vertical distance between the center of gravity (G) and the metacenter (M). It is calculated as GM = KM minus KG, where KM (keel to metacenter) comes from the vessel's stability booklet for the current displacement, and KG (keel to center of gravity) is calculated from the loading condition. A positive GM indicates initial stability; a negative GM means the vessel will capsize. The exam frequently provides KM and asks you to calculate KG from a loaded condition, then determine GM.

Q: What is free surface effect and how is the free surface correction applied?

Free surface effect is the loss of effective GM caused by liquids in partially filled tanks. When the vessel heels, liquid shifts toward the low side, effectively raising G. The free surface correction (FSC) is calculated as: FSC = (density of liquid x length x breadth cubed) divided by (12 x vessel displacement x density of seawater). The corrected GM is then: GM(corrected) = GM(solid) minus FSC. The effect is greatest when tanks are approximately 50 percent full and zero when tanks are completely full or completely empty.

Q: What is the difference between loll and list, and how do you correct each?

List is a permanent lean caused by an off-center weight (asymmetric cargo, uneven fuel burn). G is displaced to one side and the vessel rests at the angle where the righting moment equals zero on that side. Correction: shift weight to the high side to re-center G. Loll is caused by negative GM — G is above M and the vessel has no initial stability at zero degrees, so it rolls to a larger angle where the hull geometry produces enough righting lever. Correction: lower G by adding low ballast or removing topside weight. Never pump water ballast to the high side of a lolling vessel — this reduces GM further and can cause capsize.

Q: What is the angle of vanishing stability (AVS) and why does it matter?

The angle of vanishing stability (AVS), also called the angle of downflooding or the limit of positive stability, is the angle of heel at which the GZ righting arm returns to zero on the GZ curve. Beyond AVS, the vessel has no righting moment and will capsize. A higher AVS means greater range of stability. Vessels with high freeboard and flared topsides typically have AVS values of 90 to 130 degrees or more. Vessels with low freeboard and flat-sided hulls may have AVS values below 60 degrees, which is dangerous in open water.

Q: What is the difference between static stability and dynamic stability?

Static stability is the vessel's ability to return to upright from a given angle of heel when heeled slowly by a static (steady) force. It is represented by the GZ righting arm at each angle. Dynamic stability is the ability to withstand sudden or wave-induced heeling — it accounts for the energy of the heeling motion. Dynamic stability is represented by the area under the GZ curve; a larger area means greater ability to absorb wave energy without capsizing. A vessel can have a positive GZ at a given angle (static stability) but still capsize dynamically if the wave energy exceeds the area under the curve.

Q: What are LCG, TCG, and VCG and how are they used in loading calculations?

LCG (longitudinal center of gravity) is the fore-and-aft position of G measured from a reference point (usually midships or the aft perpendicular). TCG (transverse center of gravity) is the athwartship position of G from centerline. VCG (vertical center of gravity) is the height of G above the keel (also written KG). All three are calculated using the moment method: moment equals weight times arm (distance from reference). Final LCG, TCG, and VCG are found by dividing total moment by total displacement. These values determine trim (fore-aft balance from LCG), list (athwartship lean from TCG), and initial stability (from VCG via GM = KM minus KG).

Q: What is reserve buoyancy and why is the downflooding angle important?

Reserve buoyancy is the buoyant volume of the vessel above the waterline — the hull volume that provides additional flotation if the vessel is heeled or swamped. A vessel with high freeboard has high reserve buoyancy. The downflooding angle is the angle of heel at which water begins to enter an opening that cannot be sealed (a hatch, vent, or doorway). Once water enters, it adds weight and free surface effect, accelerating capsizing. USCG stability criteria require that the downflooding angle occur well before the angle of maximum righting arm. Small vessels with low cockpit sills or open hatches are especially vulnerable.

A complete reference for USCG captain license candidates. From Archimedes' principle and metacentric height through GZ curves, free surface corrections, and practical loading calculations — everything you need to master the stability and loading questions on the exam.

Key Stability Relationships

GM = KM − KGMetacentric height

KM = KB + BMKM from hull tables

GM(corrected) = GM − FSCFree surface correction

RM = Displacement × GZRighting moment

GZ = GM × sin(θ) [small angles]Righting arm approximation

Moment = Weight × ArmLoading moment calculation

1. Buoyancy and Archimedes' Principle

All vessel stability begins with Archimedes' principle: a floating object displaces a weight of fluid equal to its own weight. For a vessel, this means the weight of seawater displaced by the submerged hull exactly equals the vessel's total displacement. This upward buoyant force acts through a single point called the center of buoyancy (B).

Center of Buoyancy (B)

B is the geometric centroid of the underwater volume of the hull. Buoyancy always acts vertically upward through B. Unlike the center of gravity, B is not fixed — it shifts as the vessel heels or changes trim because the shape of the underwater volume changes. When the vessel heels to starboard, the submerged volume becomes asymmetric and B moves to starboard. This shift of B is the source of the righting force that returns the vessel to upright.

Displacement and Tons Per Inch (TPI)

Displacement is the total weight of the vessel expressed in long tons (LT) or metric tons. For exam purposes: 1 long ton equals 2,240 lb. Seawater weighs 64 lb per cubic foot (or 35 cubic feet per long ton). Fresh water weighs 62.4 lb per cubic foot. Moving from salt water to fresh water, a vessel sinks deeper because fresh water is less dense — this is the basis for the dock water allowance (DWA) used in commercial loading.

Reserve Buoyancy

Reserve buoyancy is the watertight hull volume above the current waterline. High reserve buoyancy (high freeboard, sealed deck, watertight superstructure) means the vessel can be heeled significantly before water ingress threatens the hull. Low reserve buoyancy (heavily loaded, low freeboard, open cockpit) dramatically reduces the safety margin. Freeboard is the vertical distance from the waterline to the lowest point where flooding could occur — maintaining adequate freeboard is a legal and safety requirement.

Exam Focus

Questions about reserve buoyancy typically ask: what happens to reserve buoyancy as you add cargo? (It decreases — the waterline rises.) What reduces reserve buoyancy fastest? (Overloading, or flooding a compartment.) A vessel with zero reserve buoyancy is at the point of sinking — any additional weight or downward force will cause it to sink.

2. Center of Gravity, Metacenter, and Metacentric Height (GM)

Center of Gravity (G)

G is the single point through which the total weight of the vessel acts downward. G is determined entirely by how the vessel is loaded — the weight and position of every item on board. When weight is added above the current G, G rises. When weight is added below G, G lowers. Weight added at the same height as G has no effect on G's vertical position. Critically, G does NOT move when the vessel heels — it is fixed by the loading condition. Only changing the loading can change G.

The vertical position of G above the keel is called KG. The exam frequently asks you to calculate the final KG after adding or removing weights, using the moment method: new KG equals total vertical moment divided by total displacement.

The Metacenter (M)

When a vessel is heeled to a small angle, the underwater volume shifts and B moves to the low side. A vertical line drawn upward through the new position of B intersects the vessel's original centerline at a point called the metacenter (M). For small angles of heel (under about 10-15 degrees), M is approximately fixed in position and is determined by the hull geometry — specifically by the beam and shape of the waterplane. The distance from the keel to M is called KM, and values of KM at various displacements are tabulated in the vessel's stability booklet.

Metacentric Height (GM) — The Stability Index

GM is the distance from G to M. Since KM is fixed by hull geometry and KG is determined by loading:

GM = KM − KG

Positive GM

M is above G. Vessel is stable — when heeled, buoyancy acts to return it upright. Higher GM = stiffer, faster roll.

Zero GM

M and G coincide. Vessel is in neutral equilibrium — it will stay at any angle of heel without self-righting. Extremely dangerous.

Negative GM

G is above M. Vessel is unstable — it will not return to upright when heeled and will capsize unless G is lowered immediately.

Stiff vs. Tender Vessels

A vessel with high GM is called "stiff" — it has a rapid, snappy roll period and quickly returns to upright after heeling. While stiffness sounds desirable, excessive GM causes violent rolling that is uncomfortable for crew and passengers and stresses the hull, cargo, and gear. A vessel with low (but positive) GM is "tender" — it rolls slowly and lethargically. Tender vessels feel unstable but may actually have excellent range of stability (good GZ at large angles). The ideal GM depends on the vessel type, intended service, and sea conditions.

Worked Example: GM Calculation

Vessel displacement: 45 long tons

KM from stability booklet at 45 LT: 4.20 ft

Lightship KG: 3.10 ft (from stability letter)

Crew and gear added: 2 LT at VCG 5.50 ft

── Moment method for new KG ──

Lightship moment: 43 LT × 3.10 ft = 133.3 ft-LT

Added weight moment: 2 LT × 5.50 ft = 11.0 ft-LT

Total moment: 144.3 ft-LT

New KG: 144.3 / 45 = 3.21 ft

GM = KM − KG = 4.20 − 3.21 = 0.99 ft (positive, stable)

3. Righting Arm (GZ), Righting Moment, and the Static Stability Curve

The Righting Arm (GZ)

At any given angle of heel, the horizontal distance between the line of action of buoyancy (upward through B) and the line of action of gravity (downward through G) is called the righting arm, abbreviated GZ. GZ represents the "lever" that produces the righting moment. When GZ is positive, the vessel is self-righting at that angle. When GZ is zero, the vessel is at an angle of equilibrium. When GZ becomes negative, the vessel will capsize at that angle.

For small angles of heel (under about 10-15 degrees), GZ can be approximated as:

GZ ≈ GM × sin(θ)

At larger angles, GZ diverges from this approximation because the hull shape — not just GM — dominates the righting lever. This is why GZ is plotted as a curve across all angles of heel, not calculated from GM alone at large angles.

The Righting Moment

The righting moment (RM) is the actual torque tending to return the vessel to upright:

RM = Displacement × GZ

The Static Stability Curve (GZ Curve)

The GZ curve plots righting arm (GZ, in feet or meters) against angle of heel (in degrees) for a given loading condition. Every stability booklet contains a family of GZ curves at different displacements. Key features to read from the GZ curve:

Initial slope: The slope of the GZ curve at 0 degrees equals GM. A steeper slope means higher GM. This is the initial stability region.

Maximum GZ: The peak of the GZ curve — the angle at which the vessel has maximum righting arm. Occurs typically between 30 and 60 degrees on most hull forms.

Range of positive stability: The range of angles over which GZ is positive. The wider the range, the more stable the vessel in extreme conditions.

Angle of vanishing stability (AVS): The angle where GZ returns to zero and then goes negative. Beyond AVS, the vessel capsizes. Higher AVS = safer vessel in beam seas.

Area under the curve: The area under the GZ curve represents dynamic stability — the vessel's ability to absorb wave energy. Greater area = greater ability to survive breaking waves.

Dynamic Stability vs. Static Stability

Static stability describes the vessel's behavior under slowly applied, steady heeling forces. Dynamic stability describes its behavior under sudden, wave-induced heeling where momentum and energy are involved. A wave that heels a vessel to 40 degrees and then recedes does not just create a static 40-degree heel — it imparts kinetic energy that can carry the vessel beyond 40 degrees before the righting moment can arrest the motion.

The area under the GZ curve between two angles represents the energy available to resist heeling between those angles. USCG stability criteria for uninspected passenger vessels (46 CFR 26) and small passenger vessels (46 CFR 178) specify minimum areas under the GZ curve to ensure adequate dynamic stability.

4. Free Surface Effect, Free Surface Correction, and Liquid Loading

Why Free Liquid Destroys Stability

When a tank is partially filled, the liquid inside has a free surface — a horizontal surface that can slosh. When the vessel heels, the liquid immediately flows to the low side, shifting the weight of the liquid outboard. This acts identically to raising the center of gravity: it reduces effective GM. The vessel behaves as if G were higher than it actually is, even though no weight has been added or removed.

The free surface effect depends on the breadth of the tank and the density of the liquid — not on how much liquid is in the tank (above a small threshold). A wide, shallow tank half-full produces nearly the same free surface moment as the same tank mostly full or mostly empty.

Free Surface Correction (FSC) Formula

The free surface correction quantifies the virtual rise in G caused by a slack tank. The formula for one rectangular tank:

FSC = (ρ_liquid × l × b³) / (12 × Δ × ρ_seawater)

Where: ρ_liquid = density of liquid in the tank

l = length of the tank (fore-aft)

b = breadth of the tank (the critical dimension — raised to the 3rd power)

Δ = vessel displacement in the same units

ρ_seawater = density of seawater (64 lb/ft³ or 1.025 t/m³)

Note that b is cubed — doubling the breadth of a tank increases the free surface correction by a factor of 8. This is why longitudinal tank subdivision (adding a centerline bulkhead) is so effective: it halves the breadth, cutting the FSC to one-eighth of its original value.

Worked Example: Free Surface Correction

Tank dimensions: 10 ft long × 8 ft wide, half-full of diesel (ρ = 53 lb/ft³)

Vessel displacement: 45 LT = 45 × 2,240 = 100,800 lb

Seawater density: 64 lb/ft³

── FSC calculation ──

Numerator: 53 × 10 × (8³) = 53 × 10 × 512 = 271,360

Denominator: 12 × 100,800 × 64 = 77,414,400

FSC = 271,360 / 77,414,400 = 0.0035 ft

── If GM(solid) = 1.20 ft ──

GM(corrected) = 1.20 − 0.0035 ≈ 1.197 ft

(In this example the correction is small. With a wider tank or larger vessel heel, the effect grows significantly.)

Slack Tanks and Practical Rules

Keep tanks full or empty

A completely full tank has no free surface. A completely empty tank has no liquid to slosh. Either condition eliminates free surface effect entirely.

Avoid ~50% fill in rough conditions

Free surface moment is maximum near 50% fill for most tank shapes. Deliberately fill to 95%+ or drain to 5% or less before getting underway in a seaway.

Subdivide with centerline bulkheads

A longitudinal bulkhead halves the breadth of the liquid surface, reducing FSC by 75%. Tanks designed with centerline divisions have far less free surface effect.

Dense liquids = greater FSC

Saltwater ballast (64 lb/ft³) produces more free surface effect than diesel (53 lb/ft³). The density ratio directly multiplies the FSC.

5. Loll vs. List, and Advanced Loading Calculations

List: Off-Center Weight

A list occurs when the transverse center of gravity (TCG) is offset from the vessel's centerline. When TCG is to starboard, G is to starboard of centerline, and the vessel leans to starboard until the center of buoyancy is directly below G. This is a condition of static equilibrium — the vessel remains at the list angle unless the loading is corrected. A list vessel still has positive GM; it is simply "set" to one side.

Correction for list: move weight from the low side to the high side, add weight on the high side, or remove weight from the low side. Calculate the required shift using:

Tan(θ) = TCG / GM

where θ is the angle of list

Loll: Negative Initial GM

Loll is fundamentally different from list. It occurs when GM is negative — G is above M. At zero degrees, the vessel is in an unstable position. Any small disturbance causes it to heel, and as it heels, the hull form changes: B shifts rapidly outboard (flared hulls) and KM rises. At some angle, KM has risen enough that M is again above G — now GM is positive and the vessel settles at the "loll angle." The vessel is not stable upright; it is stable only at the loll angle. The loll is equally valid to port or starboard — the vessel will flop from one loll angle to the other in a seaway, which is dangerous.

Critical Warning: Correcting Loll

Never pump ballast to the high side of a lolling vessel to "straighten it up." The vessel may momentarily pass through zero degrees — but at zero degrees, GM is still negative. Adding water ballast high actually raises G further, making GM more negative. The vessel may then roll violently past zero to the opposite loll angle, or capsize entirely. The only correct correction is to lower G: add ballast to the very lowest tanks, remove topside weight, or reduce free surface effect in high tanks by filling them completely.

Loading Calculations: LCG, TCG, and VCG

All loading calculations use the moment method. For each axis:

Term	Axis	Effect if off-center	Reference point
VCG (= KG)	Vertical	Affects GM = KM − KG	Keel (K)
LCG	Longitudinal	Creates trim (bow up or down)	Midships or AP
TCG	Transverse	Creates list (lean to one side)	Vessel centerline

For each axis, the final position of G is found by: dividing the sum of all moments by total displacement. A moment for each item equals its weight multiplied by its distance from the reference point (positive in one direction, negative in the other). The exam typically provides a loading table and asks for the final KG, LCG, or TCG, then asks you to calculate GM or list angle.

Worked Example: Loading Calculation for VCG

Item	Weight (LT)	VCG (ft)	Moment (ft-LT)
Lightship	43.0	3.10	133.3
Fuel (low tanks)	2.5	1.20	3.0
Fresh water	0.8	2.50	2.0
Passengers/gear	1.5	4.80	7.2
Deck cargo	0.5	7.20	3.6
TOTAL	48.3	—	149.1

KG = 149.1 / 48.3 = 3.09 ft

KM at 48.3 LT (from booklet) = 4.15 ft

GM = 4.15 − 3.09 = 1.06 ft (positive — stable)

6. Angle of Vanishing Stability, Downflooding, and Stability Criteria

Angle of Vanishing Stability (AVS)

The angle of vanishing stability is the heel angle at which the GZ righting arm returns to zero from the positive side. At angles beyond AVS, GZ is negative — the vessel has passed the point of no return and will capsize. The AVS depends primarily on hull form (beam, freeboard, flare, camber) and the loading condition (KG). Raising G reduces AVS because it reduces GZ at all angles.

Typical AVS values: monohull sailing vessels 100-130 degrees; power cruisers 60-90 degrees; flat-bottomed open skiffs 40-60 degrees. An AVS below 60 degrees is considered dangerous for any offshore service. The ISO and USCG stability criteria for small vessels typically require minimum AVS values depending on the intended service area.

Downflooding Angle

The downflooding angle is the heel angle at which water can enter the vessel through an opening that cannot be rapidly closed — an open hatch, vent, companionway, or low freeboard point. Once water enters, it adds weight and free surface effect, dramatically accelerating the capsizing process. USCG regulations require that the downflooding angle must be greater than the angle of maximum righting arm for vessels subject to stability regulations.

For small uninspected vessels, the captain must assess downflooding angle by inspection: identify the lowest point where water could enter in a knockdown, measure its height above the waterline, and calculate the angle at which that point reaches the water. Common downflooding points on sportfishing boats and charter vessels include cockpit drain capacity, engine room vents, and sliding cabin doors.

USCG Stability Requirements for Small Vessels

Small passenger vessels under 100 GT carrying 12 or fewer passengers for hire (T-boat category, 46 CFR subchapter T) and small passenger vessels (subchapter K) must meet specific stability criteria. For uninspected passenger vessels (OUPV/6-pack), no formal stability analysis is required by regulation, but the captain bears full responsibility for operating the vessel within safe limits.

Stability booklet

Required for inspected vessels. Contains KM at various displacements, lightship KG, maximum allowable KG, and GZ curves for key loading conditions.

Stability letter

A simplified alternative for vessels under 79 ft. States maximum allowable KG (or minimum GM) at various displacements. Simpler than a full booklet but legally binding.

Load line certificate

Required for larger vessels. Specifies maximum draft (and thus minimum freeboard) for various zones and seasons. Exceeding load line marks is a serious violation.

OUPV practical assessment

Even without formal stability documents, OUPV captains must demonstrate knowledge of: how to assess list, how loading decisions affect GM, and when to refuse to get underway.

7. Practical Stability Assessment for Small Vessel Captains

Theory becomes practice on the dock before departure. A competent captain performs a mental stability assessment before every trip, adjusting loading and briefing crew and passengers accordingly. The following principles govern practical stability management:

Know your vessel's stability letter or booklet limits

Before departure, verify the loaded displacement does not exceed the vessel's limits. If you have a stability letter specifying maximum KG at 15 LT, and you are loading 14 LT with passengers high on the flybridge, calculate whether KG is within limits.

Conduct a pre-departure stability walk-around

Check that the vessel is sitting level (no list). Verify tank levels — avoid departing with tanks near 50% in rough conditions. Confirm all hatches and vents are secured. Look for any unusual weight high on deck (diving gear, coolers, generators).

Brief passengers on stability behavior

Tell passengers to sit low, not to congregate on one rail, and not to stand on gunwales. On a 26-foot vessel, 6 passengers moving from center to the port rail can create 4-6 degrees of list. On a small vessel this is significant and potentially dangerous in a seaway.

Monitor fuel burn and tank status underway

As fuel is consumed, tanks approach the 50% range where free surface effect is worst. Plan refueling to avoid extended operation with slack tanks. Alternatively, cross-flood tanks to keep one full and one empty rather than both at 50%.

Recognize danger signs immediately

A vessel that develops a sudden list underway should be treated as an emergency until cause is identified. Sudden list can indicate: flooding of a compartment, shift of cargo, liquid ingress in a void. Reduce speed, investigate immediately, and prepare to abandon ship if flooding is suspected.

Never exceed maximum passenger or load limits

Passenger and weight limits on the Certificate of Inspection (COI) are derived from stability calculations. Exceeding them invalidates the stability analysis that supports the COI. Operating overloaded is illegal, uninsured, and potentially fatal.

Stability in Heavy Weather

In a seaway, static stability calculations are only a starting point. Wave-induced heeling can push vessels well beyond their static list angles. Key heavy-weather stability practices:

Reduce speed in beam seas

Speed generates momentum that can amplify wave-induced rolling. Slowing down reduces the energy of each wave impact.

Alter course to reduce roll

Quarter seas (waves at 45 degrees on the stern quarter) typically produce less dangerous rolling than pure beam seas. If rolling is severe, consider altering course.

Fill or empty tanks before the storm hits

Once in heavy weather it may be too late to safely manage tanks. Assess tank levels and plan fuel management before departing into forecast rough conditions.

Frequently Asked Questions

What is the metacentric height (GM) and how is it calculated?+

GM is the vertical distance from G to M. It equals KM minus KG, where KM comes from the stability booklet at the current displacement and KG is calculated from the loading condition using the moment method. Positive GM means stable; negative GM means the vessel will capsize.

What is free surface effect and how is the free surface correction applied?+

Free surface effect is the virtual rise in G caused by liquid in a partially filled tank. The correction (FSC) equals (liquid density times length times breadth cubed) divided by (12 times displacement times seawater density). The corrected GM equals GM(solid) minus FSC. The effect is worst at ~50% fill and zero when tanks are completely full or empty.

What is the difference between loll and list, and how do you correct each?+

List is caused by off-center G (TCG off centerline) — correct by moving weight to the high side. Loll is caused by negative GM — the vessel has no upright stability and rests at a larger heel angle where the hull geometry restores positive GM. Correct loll by lowering G (add low ballast, remove topside weight). Never pump ballast to the high side of a lolling vessel — it can cause capsizing.

What is the angle of vanishing stability (AVS) and why does it matter?+

AVS is the angle where the GZ righting arm reaches zero from the positive side on the GZ curve. Beyond AVS, GZ is negative and the vessel capsizes. Higher AVS means a greater range of positive stability. Raising G reduces AVS. A minimum AVS is required for vessels operating offshore or in exposed waters.

What is the difference between static stability and dynamic stability?+

Static stability is the righting moment at a given angle under a steady heeling force, represented by GZ. Dynamic stability is the energy available to resist heeling — it equals the area under the GZ curve. A vessel can have positive static stability (positive GZ) at a given angle but still capsize dynamically if an incoming wave imparts more energy than the area under the GZ curve can absorb.

What are LCG, TCG, and VCG and how are they used in loading calculations?+

VCG (equals KG) is the height of G above the keel — it determines GM. LCG is the fore-aft position of G — it determines trim. TCG is the athwartship position of G — it determines list. All three are calculated using the moment method: sum of (weight times arm) divided by total displacement. The exam provides a loading table and asks you to find the final G position in one or more of these axes.

What is reserve buoyancy and why is the downflooding angle important?+

Reserve buoyancy is the watertight hull volume above the current waterline that provides additional flotation when heeled. The downflooding angle is the heel angle at which water enters through an unsealed opening. Once water ingress begins, free surface effect and added weight accelerate capsizing. Reserve buoyancy and a high downflooding angle are critical safety margins for small vessels operating in a seaway.

Exam Quick Reference

Adding weight ABOVE G

Raises G, reduces GM, reduces stability, reduces AVS

Adding weight BELOW G

Lowers G, increases GM, improves stability, increases AVS

Filling a tank from slack to full

Eliminates free surface effect, may raise or lower G depending on tank position

Tank at ~50% full

Maximum free surface effect — worst case for stability

G above M (negative GM)

Vessel is unstable at zero degrees — loll or capsize

High freeboard, sealed deck

High reserve buoyancy, high downflooding angle, larger range of stability

Passengers crowding one rail

Moves TCG off centerline, creates list, raises effective G — double hazard

Wide tank vs. narrow tank (same volume)

Wider tank has far greater free surface effect (breadth is cubed in FSC formula)

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