
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
\(ABCD\) is a convex quadrilateral and \(E, F\) are the mid-points of \(BC\) and \(CD\) respectively. The line segments \(AE, AF\) and \(EF\) divide \(ABCD\) into four triangles, whose areas are four consecutive integers. Find the greatest possible area of \(\triangle ABD\).

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