
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
Let  $A$ ,  $B$ ,  $C$ , and  $P$  be points in the plane such that no three of them are collinear. Suppose that the areas of triangles  $BPC$ ,  $CPA$ , and  $APB$  are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle  $ABC$ .

*Proposed by Ankan Bhattacharya*

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