
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 triangle  $ ABC$ ,  $ AB \equal{} 10$ ,  $ BC \equal{} 14$ , and  $ CA \equal{} 16$ . Let  $ D$  be a point in the interior of  $ \overline{BC}$ . Let  $ I_B$  and  $ I_C$  denote the incenters of triangles  $ ABD$  and  $ ACD$ , respectively. The circumcircles of triangles  $ BI_BD$  and  $ CI_CD$  meet at distinct points  $ P$  and  $ D$ . The maximum possible area of  $ \triangle BPC$  can be expressed in the form  $ a\minus{}b\sqrt{c}$ , where  $ a$ ,  $ b$ , and  $ c$  are positive integers and  $ c$  is not divisible by the square of any prime. Find  $ a\plus{}b\plus{}c$ .

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