
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
Let  $a_1$ ,  $a_2$ ,  $\cdots$  be a sequence such that  $a_1=a_2=\frac 15$ , and for  $n \ge 3$ ,   $$ a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}. $$   Find the smallest integer  $n$  such that  $a_n>1-5^{-2022}$ .

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