
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
Given the ellipse \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\), a line passing through its left focus \( F_{1} \) intersects the ellipse at points \( A \) and \( B \). Point \( D(a, 0) \) is located to the right of \( F_{1} \). Lines \( A D \) and \( B D \) intersect the ellipse's left directrix at \( M \) and \( N \), respectively. If the circle with diameter \( M N \) passes exactly through \( F_{1} \), find the value of \( a \).

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