
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
Compute the triple integral

$$
\iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} \, dx \, dy \, dz
$$

where the region $\Omega$ is bounded by the surfaces

$$
z = \sqrt{36 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{3}}
$$

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