
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 $P(n)$ denote the product of the digits of the number $n$. For example, $P(58) = 5 \times 8 = 40$ and $P(319) = 3 \times 1 \times 9 = 27$.

(a) What are the natural numbers less than 1000 whose product of their digits is 12, i.e., the natural numbers $n < 1000$ such that $P(n) = 12$?

(b) How many natural numbers less than 199 satisfy $P(n) = 0$? That is, have the product of their digits equal to 0?

(c) What natural numbers less than 200 satisfy the inequality $37 < P(n) < 45$?

(d) Among the numbers from 1 to 250, which number has the greatest product of its digits?

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