
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
A positive integer \( N \) is piquant if there exists a positive integer \( m \) such that if \( n_{i} \) denotes the number of digits in \( m^{i} \) (in base 10), then \( n_{1} + n_{2} + \cdots + n_{10} = N \). Let \( p_{M} \) denote the fraction of the first \( M \) positive integers that are piquant. Find \(\lim_{M \rightarrow \infty} p_{M}\).

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