
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
In the convex quadrilateral \(ABCD\), the midpoints of sides \(BC\) and \(CD\) are \(E\) and \(F\) respectively. The segments \(AE\), \(EF\), and \(AF\) divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle \(ABD\)?

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