
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 its diagonal, one of the parts is flipped, and the two parts are glued along the cut line to form $F_{k+1}$. 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 any movement.)

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