
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
Let  $S$  be the number of ordered pairs of integers  $(a,b)$  with  $1 \leq a \leq 100$  and  $b \geq 0$  such that the polynomial  $x^2+ax+b$  can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when  $S$  is divided by  $1000$ .

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