
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 triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 3012$, $AC = 3010$, and $BC = 3009$, find the length of $PQ$. Express $PQ$ as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers, and find $m + n$.

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