
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 inscribed in a circle with shortest side \(AB\). The ratio \(\frac{[BCD]}{[ABD]}\) is an integer (where \([XYZ]\) denotes the area of triangle \(XYZ\)). If the lengths of \(AB, BC, CD\), and \(DA\) are distinct integers no greater than 10, find the largest possible value of \(AB\).

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