
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
If a sequence $\{a_{n}\}$ with $n$ terms satisfies: $a_{i}=a_{n+1-i}$ for $i=1,2,3,\ldots ,n$, we call it an "$n$-term symmetric sequence". For example, the sequence $1,2,2,1$ is a $4$-term symmetric sequence; the sequence $1,2,3,2,1$ is a $5$-term symmetric sequence. Let $\{c_{n}\}$ be a $2k+1$-term symmetric sequence, where $c_{1},c_{2},\ldots,c_{k+1}$ form an arithmetic sequence with a common difference of $-2$, and the smallest term of the sequence $\{c_{n}\}$ is $-10$. Let the sum of the first $2k+1$ terms of the sequence $\{c_{n}\}$ be $S_{2k+1}$. If $S_{2k+1}=-50$, then the value of $k$ is ______.

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