
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 solid $\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\alpha$, and $\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\angle A=90^{\circ}$. Point $P$ is a moving point on the solid $\Omega$, and the projection of $P$ on the circle $O'$ lies on the circumference of $O'$. Given $OO'=1$, the maximum volume of the tetrahedron $P-ABC$ is \_\_\_\_\_\_

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