
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 $S(n) = \tau(1) + \tau(2) + \ldots + \tau(n).$ Find the absolute difference $|a-b|$ where $a$ denotes the number of integers $n \leq 3000$ for which $S(n)$ is odd, and $b$ the number of integers $n \leq 3000$ for which $S(n)$ is even.

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