
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 solid $\Omega$ is bounded by the planes $x=2$, $y=0$, $y=1$, $z=0$ and the cylinder $z^2 = 6x$. The volume density of the material at each point is proportional to the distance from the $XOY$ plane. Find the moment of inertia of the solid $\Omega$ relative to the $z$-axis.

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