
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

In a vertical stationary cylindrical vessel closed by a piston of mass \( M \) is an ideal gas. The gas is heated, causing the piston to move from rest with a constant acceleration \( a \). During the heating period, the piston exits the vessel and then continues to move vertically upwards in free space. The average power of the heater is \( P \). Determine the maximum height the piston reaches above the edge of the vessel. The molar heat capacity of the gas at constant volume is \( c_{v} \). The acceleration due to gravity is \( g \). Neglect the heat capacity of the vessel and the piston. There is a vacuum on the external side of the piston.

Present the answer in LaTex format: \boxed{Your answer}
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