
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  $N$  be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to  $N$ , independently and uniformly at random. Let  $p_N$  denote the probability that the product of these two integers has a units digit of  $0$ . The maximum possible value of  $p_N$  over all possible choices of  $N$  can be written as  $\tfrac ab,$  where  $a$  and  $b$  are relatively prime positive integers. Compute  $100a + b$ .

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