
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
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. Let $A(a_{1009},1)$, $B(2,-1)$, and $C(2,2)$ be three points on the coordinate plane, and $O$ be the origin. If the projections of vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ in the direction of vector $\overrightarrow{OC}$ are the same, determine the value of $S_{2017}$.

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