
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 number of increasing sequences of positive integers \( a_1 \le a_2 \le a_3 \le \cdots \le a_{12} \le 2030 \) such that \( a_i - i \) is a multiple of 4 for \( 1 \le i \le 12 \), can be expressed as \( {m \choose n} \) for some positive integers \( m > n \). Compute the remainder when \( m \) is divided by 1000.

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