
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 \( ABC \) be a triangle with \( AB = AC = \frac{25}{14} BC \). Let \(M\) denote the midpoint of \(\overline{BC}\) and let \(X\) and \(Y\) denote the projections of \(M\) onto \(\overline{AB}\) and \(\overline{AC}\), respectively. If the areas of triangle \(ABC\) and quadrilateral \(AXMY\) are both positive integers, find the minimum possible sum of these areas.

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