
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 a point chosen on the interior of side  $\overline{BC}$  of triangle  $\triangle ABC$  with side lengths  $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$ . If  $X$  and  $Y$  are the feet of the perpendiculars from  $P$  to the sides  $AB$  and  $AC$ , then the minimum possible value of  $PX^2 + PY^2$  can be expressed as  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by Andrew Wen*

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