
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

Consider the equilibrium of gases in communicating vessels. Let there be two vessels with volumes $V_{1}$ and $V_{2}$ respectively, filled with gas. The initial gas pressure is $P_{1}$ in the first vessel and $P_{2}$ in the second. The vessels are connected by a tube through which gas flows from one vessel to the other. Assuming that the amount of gas flowing per second is proportional to the difference of the squares of the pressures, determine the pressures $p_{1}$ and $p_{2}$ in the vessels at time $t$.

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