
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 trapezoid $ABCD$ with bases $AB = a$ and $CD = b$, a segment $A_1B_1$ is drawn connecting the midpoints of the diagonals. In the resulting trapezoid, another segment $A_2B_2$ is drawn, also connecting the midpoints of the diagonals, and so on. Can any number in the sequence of lengths $AB, A_1B_1, A_2B_2, \ldots$ appear twice? Is this sequence monotonic (increasing or decreasing)? Does it converge to any limit?

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