
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 an acute triangle with \(A\)-excircle \(\Gamma\). Let the line through \(A\) perpendicular to \(BC\) intersect \(BC\) at \(D\) and intersect \(\Gamma\) at \(E\) and \(F\). Suppose that \(AD = DE = EF\). If the maximum value of \(\sin B\) can be expressed as \(\frac{\sqrt{a} + \sqrt{b}}{c}\) for positive integers \(a, b\), and \(c\), compute the minimum possible value of \(a + b + c\).

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