
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 a natural number  $b$ , let  $N(b)$  denote the number of natural numbers  $a$  for which the equation  $x^2 + ax + b = 0$  has integer roots. What is the smallest value of  $b$  for which  $N(b) = 20$ ?

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