
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
The sequence of real numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{100}\) satisfies the inequalities \(a_{n} - 2022 a_{n+1} + 2021 a_{n+2} \geq 0\) for \(n = 1, 2, 3, \ldots, 98\), and \(a_{99} - 2022 a_{100} + 2021 a_{1} \geq 0\), \(a_{100} - 2022 a_{1} + 2021 a_{2} \geq 0\). Find \(a_{22}\), given that \(a_{10} = 10\).

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