
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  $P$  be an interior point of triangle  $ABC$ . Let  $a,b,c$  be the sidelengths of triangle  $ABC$  and let  $p$  be it's semiperimeter. Find the maximum possible value of  $$  \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$  taking into consideration all possible choices of triangle  $ABC$  and of point  $P$ .

by Elton Bojaxhiu, Albania

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