
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
A sequence is defined as follows: $a_1 = 2, a_2 = 1, a_3 = 3$, and for all positive integers $n$, $a_{n+3} = 2(a_{n+2} + a_{n+1} + a_n)$. Given $a_{20} = 797161$, $a_{21} = 1812441$, and $a_{22} = 4126643$, find the remainder when $\displaystyle \sum_{k=1}^{19}a_k$ is divided by $1000$.

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