
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
Suppose tetrahedron  $PABC$  has volume  $420$  and satisfies  $AB = 13$ ,  $BC = 14$ , and  $CA = 15$ . The minimum possible surface area of  $PABC$  can be written as  $m+n\sqrt{k}$ , where  $m,n,k$  are positive integers and  $k$  is not divisible by the square of any prime. Compute  $m+n+k$ .

*Ray Li*

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