
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
Consider a positive real number $x > 1$, and the Riemann zeta function $\zeta(x)$ defined by
\[\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}.\]
Compute the sum
\[\sum_{k=2}^\infty \left\{\zeta\left(\frac{2k}{k+1}\right)\right\}.\]
Note: For a real number $x$, $\{x\}$ denotes the fractional part of $x$.

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