
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
Let \( K(x) \) be the number of irreducible fractions \( \frac{a}{b} \) such that \( a < x \) and \( b < x \) (where \( a \) and \( b \) are natural numbers).

For example, \( K\left(\frac{5}{2}\right) = 3 \) (the fractions are \( 1, 2, \frac{1}{2} \)).

Compute the sum \( K(100) + K\left(\frac{100}{2}\right) + K\left(\frac{100}{3}\right) + \ldots + K\left(\frac{100}{99}\right) + K\left(\frac{100}{100}\right) \).

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