
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 \(a_1, a_2, \ldots\) be a sequence of integers defined by \(a_1 = 3\), \(a_2 = 3\), and
\[a_{n+2} = a_{n+1} \cdot a_n - a_{n+1} - a_n + 2\]
for all \(n \geq 1\). Find the remainder when \(a_{2020}\) is divided by 22.

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