
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 $f(x)=\sin x+\tan x$. A 31-term arithmetic sequence $\{a_{n}\}$ satisfies $a_{n}\in(-\frac{\pi}{2},\frac{\pi}{2})$, and the common difference $d\neq 0$. If $f(a_{1})+f(a_{2})+...+f(a_{31})=0$, find the value of $k$ such that $f(a_{k})=0$.

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