
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 the function $f(x) = x + \sin x$. There is an arithmetic sequence $\{a_n\}$ which has 19 terms and satisfies $a_n \in \left(- \frac{\pi}{2}, \frac{\pi}{2}\right)$, with a common difference $d \neq 0$. If $f(a_1) + f(a_2) + \ldots + f(a_{18}) + f(a_{19}) = 0$, then $f(a_k) = 0$ when $k = \_\_\_\_\_\_ $.

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