
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 triangular pyramid $P-ABC$ with $PA$, $PB$, and $PC$ being mutually perpendicular to each other, and $PA = PB = 2PC = 2a$. If the surface area of the sphere circumscribing the pyramid $P-ABC$ is $S = 9\pi$, then the value of the real number $a$ is ___.

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