
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
For a positive real number \( x > 2 \), the modified Riemann zeta function \( \xi(x) \) is defined by
\[
\xi(x) = \sum_{n=1}^\infty \frac{1}{(2n)^x}.
\]
Compute
\[
\sum_{k=3}^\infty \{\xi(2k+1)\}.
\]
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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