
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
In triangle $ABC$, where $AB=13, BC=14,$ and $CA=15.$ Circle $\omega_1$ passes through $C$ and is tangent to line $AB$ at $A.$ Circle $\omega_2$ passes through $B$ and is tangent to line $AC$ at $A.$ Let $M$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Determine the length of $AM=\tfrac pq,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

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