
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
Given the schematic of an air compressor, we want to achieve a pressure \( p \) in a tank with volume \( V_{2} \). The piston, which has a volume \( V_{1} \) and moves frictionlessly in a cylinder with a constant cross-sectional area, is initially at atmospheric pressure and temperature. How much mechanical work is needed to achieve the pressure \( p \) in the tank? (Assume the processes are isothermal and no work is done while pulling the piston.)

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