
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 \(ABCD\) be a convex quadrilateral and \(E\) be a point on \(CD\) such that the circumcircle of \(\triangle ABE\) is tangent to \(CD\). Suppose \(AC\) meets \(BE\) at \(F\), \(BD\) meets \(AE\) at \(G\), and \(AC\) meets \(BD\) at \(H\). If \(FG \parallel CD\), and the areas of \(\triangle ABH\), \(\triangle BCE\), and \(\triangle ADE\) are 2, 3, and 4 respectively, find the area of \(\triangle ABE\).

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