
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 \( \frac{a}{b} \), 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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