
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  $n$ . For example,  $\tau (1)=1$  and  $\tau(6) =4$ . Define  $S(n)$  by \[S(n)=\tau(1)+ \tau(2) + ... + \tau(n).\] Let  $a$  denote the number of positive integers  $n \leq 2005$  with  $S(n)$  odd, and let  $b$  denote the number of positive integers  $n \leq 2005$  with  $S(n)$  even. Find  $|a-b|$ .

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