
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 that the function $f(x)$ is a quadratic function, the graph of $f(x)$ passes through the point $(0,2)$, with the axis of symmetry at $x=-\frac{1}{2}$, and the minimum value of $f(x)$ on $R$ is $\frac{7}{4}$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ When $x\in [m-2,m]$, $m\in R$, find the minimum value of the function $f(x)$ (expressed in terms of $m$);  
$(3)$ If the function $F(x)=f(x)-ax$ has only one zero in the interval $(0,3)$, find the range of values for $a$.

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