
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 modified sequence is defined as follows: \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 3\), and for all positive integers \(n\), \(a_{n+3} = a_{n+2} + a_{n+1} + a_n\). Given \(a_{15} = 3136\), \(a_{16} = 5768\), and \(a_{17} = 10609\), find the remainder when \(\displaystyle \sum_{k=1}^{15}a_k\) is divided by 1500.

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