
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 a pyramid with a square base of side length $\sqrt {2}$ and lateral edges of length $\sqrt {5}$. A cylinder has one of its circular bases passing through the midpoints of the four lateral edges of the pyramid, while its other circular base has its center at the center of the pyramid's base. Calculate the volume of the cylinder.

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