
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 the trapezoid \(ABCD\) with bases \(AB = a\) and \(CD = b\), segment \(A_1B_1\) is drawn, connecting the midpoints of the diagonals. In the resulting trapezoid, segment \(A_2B_2\) is similarly 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}
user