
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 regular quadrilateral pyramid PABCD (where $P$ is the apex) with base side $a$ and lateral edge $a$. A sphere centered at point $O$ passes through point $A$ and touches the edges $PB$ and $PD$ at their midpoints. Find the volume of the pyramid $OPCD$.

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