
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
A sphere is inscribed in a regular quadrilateral pyramid, touching the base and all the lateral faces. The sphere divides the height of the pyramid in a ratio of $9:7$ from the apex of the pyramid. Find the volume of the pyramid if the side of the pyramid's base is $a$.

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