
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
Given a finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers (with \(n \geq 3\)), and the recurrence relation \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\) holds for all \(\kappa \leq n-2\). The sequence must contain \(a_{k} = 2022\). Determine the maximum number of three-digit numbers that are multiples of 4 that this sequence can contain.

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