
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
Solve the equation
\[
\frac{1}{x+2} + \frac{1}{x+4} - \frac{1}{x+6} - \frac{1}{x+8} - \frac{1}{x+10} + \frac{1}{x+12} + \frac{1}{x+14} + \frac{1}{x+16} = 0
\]
and find the sum \(a + b + c + d\) for the roots of the form \(-a \pm \sqrt{b \pm c \sqrt{d}}\) where \(a, b, c, d\) are positive integers and \(d\) is not divisible by the square of a prime.

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