
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
In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\alpha$ through $AM$ parallel to $BC$, and denote $\angle PAM$ as $\theta_1$. The acute dihedral angle between $\alpha$ and the base plane $ABC$ is denoted as $\theta_2$. When $\theta_1$ is maximized, find the value of $\tan \theta_2$.

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