
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 length of the intervals \((c, d)\), \([c, d]\), \((c, d]\), and \([c, d)\) as \(d - c\), where \(d > c\). Given real numbers \(a > b\), find the sum of the lengths of the intervals of \(x\) that satisfy \(\frac{1}{x-a} + \frac{1}{x-b} \geq 1\).

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