
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
Given a convex quadrilateral \(ABCD\). Point \(M\) is the midpoint of side \(BC\), and point \(N\) is the midpoint of side \(CD\). The segments \(AM\), \(AN\), and \(MN\) divide the quadrilateral into four triangles whose areas, in some order, are consecutive natural numbers. What is the largest possible area that triangle \(ABD\) can have?

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