
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 the sequence $\{a_{n}\}$, where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}a_{n}+5, & \frac{n}{3}∉{N}^{*}\\ a_{n-1}, & \frac{n}{3}∈{N}^{*}\end{array}\right.$, find the maximum value of $k(k\in N^{*})$ for which $a_{n}\leqslant 2021$ holds for any $n\leqslant k$.

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