
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
When manufacturing an electronic digital computer, regardless of the base used, each digit requires a device. If a base-$n$ system is used, each digit in the system has $n$ possible values. To represent an $m$-digit number in a base-$n$ system, $m \times n$ devices are needed. Suppose the maximum number that the machine can represent is $M$ (using base-$n$, where $n \geq 2$), and the number of devices required is $G$.

1. If the machine needs to represent an $m$-digit number in base-$n$, write an equation relating $M$, $m$, and $n$.
2. Treat $G$ as a function of $n$, with $M$ as a constant, and write this functional relationship.
3. Calculate $G(2)$, $G(3)$, $G(4)$, and $G(5)$ in terms of $\lg (M+1)$.
4. Given $M$, what value of $n$ minimizes the number of devices $G$?

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