
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 $f(x) = \frac{1}{2}x^2 + \frac{b}{x} + c$ ($b, c$ are constants) and $g(x) = \frac{1}{4}x + \frac{1}{x}$ are functions defined on $M=\{x|1\leq x \leq 4\}$, find the maximum value of $f(x)$ on the set $M$.

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