
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
For positive integers $n$, let $\tau(n)$ denote the number of positive integer divisors of $n$, including 1 and $n$. Define $S(n)$ by $S(n) = \tau(1) + \tau(2) + \cdots + \tau(n)$. Let $a$ denote the number of positive integers $n \leq 1000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 1000$ with $S(n)$ even. Find $|a - b|$.

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