
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 $\triangle ABC$. Let $T$ and $U$ be the points where the circles inscribed in the triangles $ACT$ and $BCT$ are tangent to $\overline{CH}$. If $AB = 2023,$ $AC = 2022,$ and $BC = 2021,$ then $TU$ can be expressed as $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$

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