
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 \(AB = 2\), \(CA = 3\), and \(BC = 4\). Let \(D\) be the point diametrically opposite \(A\) on the circumcircle of \(\triangle ABC\), and let \(E\) lie on line \(AD\) such that \(D\) is the midpoint of \(\overline{AE}\). Line \(l\) passes through \(E\) perpendicular to \(\overline{AE}\), and \(F\) and \(G\) are the intersections of the extensions of \(\overline{AB}\) and \(\overline{AC}\) with \(l\). Compute \(FG\).

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