
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) Given that the decimal representation of \(5^{2018}\) has 1411 digits and begins with 3 (the leftmost non-zero digit is 3), for how many integers \(1 \leq n \leq 2017\) does the number \(5^{n}\) begin with 1?

b) The integers \(4^{52}\) and \(5^{52}\) both begin with the digit 2. If the decimal representations of the powers \(4^{n}\) and \(5^{n}\), with \(n > 0\) and an integer, both start with the same digit \(d\), what are the possible values of this digit?

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