
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
Find the number of increasing sequences of positive integers $b_1 \le b_2 \le b_3 \le \cdots \le b_{15} \le 2023$ such that $b_i - i$ is divisible by 3 for $1\le i \le 15$. Express your answer as ${p \choose q}$ for positive integers $p > q$, and compute the remainder when $p$ is divided by 1000.

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