
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
A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ($n \geq 3$) of natural numbers is given, such that for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must 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}
user