
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
For all positive integers \( n \) less than 1001, define:
\[
a_n = \begin{cases} 
    16, & \text{if } n \text{ is divisible by } 15 \text{ and } 16; \\
    17, & \text{if } n \text{ is divisible by } 16 \text{ and } 17; \\
    18, & \text{if } n \text{ is divisible by } 15 \text{ and } 17; \\
    0, & \text{otherwise}.
\end{cases}
\]
Calculate \(\sum_{n=1}^{1000} a_n\).
A) 169
B) 175
C) 180
D) 185

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