
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
Find the volume of the pyramid \( PABCD \), which has a quadrilateral \( ABCD \) with sides 5, 5, 10, and 10 as the base, its shorter diagonal being \( 4\sqrt{5} \), and all the lateral faces inclined at an angle of \( 45^\circ \) to the base.

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