
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 $\overline{CH}$ be an altitude of $\triangle ABC$. Let $P\,$ and $Q\,$ be the points where the circles inscribed in the triangles $ACH\,$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 2015\,$, $AC = 2013\,$, and $BC = 2012\,$, then $PQ\,$ can be expressed as $m/n\,$, where $m\,$ and $n\,$ are relatively prime integers. Find $m + n\,$.

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