
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 task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.

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