
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 \( a_{1}, a_{2}, \cdots, a_{100} \) be non-negative integers satisfying the following conditions:
(1) There exists a positive integer \( k \leq 100 \) such that \( a_{1} \leq a_{2} \leq \cdots \leq a_{k} \), and when \( i > k \), \( a_{i} = 0 \);
(2) \( a_{1} + a_{2} + a_{3} + \cdots + a_{100} = 100 \);
(3) \( a_{1} + 2a_{2} + 3a_{3} + \cdots + 100a_{100} = 2022 \).

Find the minimum possible value of \( a_{1} + 2^{2}a_{2} + 3^{2}a_{3} + \cdots + 100^{2}a_{100} \).

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