
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 \( T(n) \) as the sum of all triangular numbers up to \( n \). Define \( S(n) \) by \( S(n) = \tau(1) + \tau(2) + \cdots + \tau(n) + T(n) \). Let \( a \) denote the number of positive integers \( n \leq 3025 \) with \( S(n) \) odd, and let \( b \) denote the number of positive integers \( n \leq 3025 \) with \( S(n) \) even. Find \( |a-b| \).

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