
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
The equation
\[\frac{1}{x} + \frac{1}{x + 3} - \frac{1}{x + 5} - \frac{1}{x + 7} - \frac{1}{x + 9} - \frac{1}{x + 11} + \frac{1}{x + 13} + \frac{1}{x + 15} = 0\]
has four 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. Find $a + b + c + d$.

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