
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
Given a sequence $\left\{a_n\right\}$ with 1001 terms, where $a_1=0$, $a_{1001}=2020$, and $a_{k+1}-a_k=1$ or $3$ for $k=1,2,\ldots,1000$. Find the number of such possible sequences (in terms of binomial coefficients).

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