
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 \( A_{1}, A_{2}, \ldots, A_{n} \) be \( n \) consecutive points on a line such that \( A_{1} A_{n} = 1 \) and let a certain number \( a \) be given (where \( 0 \leq a \leq 1 \)). What is the maximum possible number of segments \( A_{i} A_{j} \) (where \( i, j = 1, 2, \ldots, n \)) with length \( \geq a \)?

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