
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 $a_{n}$ denote the number of numbers among $2^{1}, 2^{2}, \ldots, 2^{n}$ that begin with the digit 1. Calculate the limit $\lim _{n \rightarrow \infty} \frac{a_{n}}{n}$.

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