
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 quadratic function $f(x)=x^{2}+bx+c$ that satisfies $f(2)=f(-2)$, and one root of the function $f(x)$ is $1$.
(I) Find the analytic expression of the function $f(x)$;
(II) For any $x\in[\frac{1}{2},+\infty)$, the equation $4mf(x)+f(x-1)=4-4m$ has a solution. Determine the range of real values for $m$.

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