
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
Define a sequence by $b_1 = 2$, $b_2 = 3$, $b_3 = 4$, and for all positive integers $n$, $b_{n+3} = b_{n+2} + b_{n+1} + b_n$. Given $b_{25} = 1639857$, find the remainder when $\sum_{k=1}^{25} b_k$ is divided by $500$.

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