
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 $n.$ Calculate $S(n)=\tau(1)+\tau(2)+\cdots+\tau(n)$. Let $a$ be the number of positive integers $n \leq 1500$ for which $S(n)$ is odd, and let $b$ be the number of positive integers $n \leq 1500$ with $S(n)$ even. Determine $|a-b|.$

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