
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 a cyclic quadrilateral \(ABCD\), \(\angle A=60^\circ\), \(BC = AD = 1\), ray \(AB\) and ray \(CD\) intersect at \(E\), ray \(BC\) and ray \(AD\) intersect at \(F\). Given that the perimeters of triangles \(\triangle BCE\) and \(\triangle CDF\) are integers, find the perimeter of quadrilateral \(ABCD\).

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