
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 non-negative integer. Define the *decimal digit product* \(D(n)\) inductively as follows:

- If \(n\) has a single decimal digit, then let \(D(n) = n\).

- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).

Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0.

Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\).

*proposed by the ICMC Problem Committee*

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