
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
Compute the number of positive integers  $b$  where  $b \le 2013$ ,  $b \neq 17$ , and  $b \neq 18$ , such that there exists some positive integer  $N$  such that  $\dfrac{N}{17}$  is a perfect  $17$ th power,  $\dfrac{N}{18}$  is a perfect  $18$ th power, and  $\dfrac{N}{b}$  is a perfect  $b$ th power.

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