
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 \( F_{1}, F_{2}, F_{3}, \ldots \) be a sequence of convex quadrilaterals where \( F_{k+1} \) (for \( k=1, 2, 3, \ldots \)) is obtained as follows: \( F_{k} \) is cut along a diagonal, one of the parts is flipped over, and then glued along the cut line with the other part. What is the maximum number of distinct quadrilaterals that can be contained in this sequence? (Polygons are considered distinct if they cannot be made to coincide by motion.)

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