
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
Let  $n$  be a natural number. Find all real numbers  $x$  satisfying the equation   $$ \sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4. $$  

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