
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
Given the function $f(x) = x^2 - 2bx + \frac{b}{4}$, defined on $[1, 4]$, where $b \geqslant 1$,
(I) Find the minimum value of $f(x)$, denoted as $g(b)$;
(II) Find the maximum value of $g(b)$, denoted as $M$.

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