
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 \( \triangle{ABC} \) with \( AB = 10 \), \( BC = 11 \), and \( AC = 16 \), let \( M \) be a point on \( \overline{AC} \) such that the areas of \( \triangle{ABM} \) and \( \triangle{BCM} \) are equal. Determine \( \frac{AM}{MC} \) and find the sum of the numerator and denominator.

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