
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 largest value of the real number  $ \lambda$ , such that as long as point  $ P$  lies in the acute triangle  $ ABC$  satisfying  $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$ , and rays  $ AP$ ,  $ BP$ ,  $ CP$  intersect the circumcircle of triangles  $ PBC$ ,  $ PCA$ ,  $ PAB$  at points  $ A_1$ ,  $ B_1$ ,  $ C_1$  respectively, then  $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$ .

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