
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 cylinder $AB$ with a length of $2l+d$, a well-sealing piston $C$ with a thickness $d$ and weight $P$ can slide without friction. The base plates $A$ (top) and $B$ (bottom) are equipped with valves $a$ and $b$, respectively. The following experiments are conducted:

1. When $C$ is positioned equidistant from the base plates, we open the valves $a$ and $b$; then, we close the valves, and $C$ descends.

2. We then open the valve $b$.

Calculate how much $C$ descended in both cases and determine the pressure in the $AC$ region in the first case (at the end of the experiment).

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