
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 $f(x)$ whose graph passes through the point $(2,5)$, and it satisfies $f(x+1)=f(x)-2x+1$

  
$(1)$ Find the expression of the function $f(x)$  

$(2)$ Find the minimum value of $g(x)=(2-2m)x-f(x)$ when $x∈[0,2]$

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