
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 passes through its left focus \(F_{1}\) and intersects the ellipse at points \(A\) and \(B\). Point \(D(a, \theta)\) is located to the right of \(F_{1}\). The lines \(AD\) and \(BD\) intersect the left directrix of the ellipse at points \(M\) and \(N\) respectively. If the circle for which \(MN\) is the diameter passes exactly through \(F_{1}\), find the value of \(a\).

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