
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 2014 real numbers \( x_{1}, x_{2}, \cdots, x_{2014} \) that satisfy the system of equations
$$
\sum_{k=1}^{2014} \frac{x_{k}}{n+k}=\frac{1}{2 n+1} \quad \text{for} \quad n=1,2, \cdots, 2014,
$$
calculate the value of \( \sum_{k=1}^{2014} \frac{x_{k}}{2 k+1} \).

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