
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 $f(x)= \frac{1}{2}x^{2}+ \frac{b}{x}+c$ (where $b,c$ are constants) and $g(x)=\frac{1}{4}x+\frac{1}{x}$ are functions defined on $M=\{x| 1\leqslant x\leqslant 4\}$, for any $x\in M$, there exists $x_{0}\in M$ such that $f(x)\geqslant f(x_{0}),g(x)\geqslant g(x_{0}),$ and $f(x_{0})=g(x_{0})$, then the maximum value of $f(x)$ on $M$ is $\boxed{5}$.

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