
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:
\[
\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{x + 6} - \frac{1}{x + 10} + \frac{1}{x + 12} + \frac{1}{x + 16} - \frac{1}{x + 18} - \frac{1}{x + 20} = 0
\]
and find the sum \(a + b + c + d\) where the roots are of the form \(-a \pm \sqrt{b \pm c \sqrt{d}}\).

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