
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
\[\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{x + 8} - \frac{1}{x + 12} + \frac{1}{x + 16} + \frac{1}{x + 20} - \frac{1}{x + 24} - \frac{1}{x + 28} = 0\]
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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