
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 a quadratic function $f(x)$ that satisfies $f(x+1)-f(x)=-2x+1$, and $f(2)=15$. Let $g(x)=(1-2m)x-f(x)$.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ If the function $g(x)$ is not monotonic on the interval $[0,2]$, find the range of real numbers for $m$.  
$(3)$ Find the minimum value of the function $g(x)$ on the interval $[0,2]$.

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