
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
The sequence \( u_0, u_1, u_2, \ldots \) is defined as follows: \( u_0 = 0 \), \( u_1 = 1 \), and \( u_{n+1} \) is the smallest integer \( > u_n \) such that there is no arithmetic progression \( u_i, u_j, u_{n+1} \) with \( i < j < n+1 \). Find \( u_{100} \).

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