
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
Find the maximum value of the real number \(\lambda\). Such that, for any point \(P\) inside the acute triangle \(\triangle ABC\), with \(\angle PAB = \angle PBC = \angle PCA\), and the rays \(AP, BP, CP\) intersecting the circumcircles of \(\triangle PBC, \triangle PCA, \triangle PAB\) at points \(A_1, B_1, C_1\) respectively, the expression \(S_{\triangle PBC} + S_{\triangle B_1CA} + S_{\triangle C_1AB}\) satisfies a relation to \(\lambda S_{\triangle ABC}\).

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