
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
Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \in \mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality
\[
\frac{1}{x-20}+\frac{1}{x-17} \geqslant \frac{1}{512}
\]

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