
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 $a_n$ as follows:
$$
a_n = \begin{cases} 
10, & \text{if } n \text{ is divisible by both } 13 \text{ and } 14;\\
12, & \text{if } n \text{ is divisible by both } 14 \text{ and } 11;\\
15, & \text{if } n \text{ is divisible by both } 11 \text{ and } 13;\\
0, & \text{otherwise}.
\end{cases}
$$
Calculate $\sum_{n=1}^{1000} a_n$.
A) 220 
B) 227
C) 234
D) 250

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