
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 $(0,4)$, and has two real roots $x\_1=1$ and $x\_2=4$ for the equation $f(x)=2x$. The function $g(x)=2x+m$ is also defined.

(1) Find the analytical expression of $f(x)$;
(2) If the function $h(x)=f(x)-(2t-3)x (t\in\mathbb{R})$ has a minimum value of $\frac{7}{2}$ in the interval $x\in[0,1]$, find the value of $t$;
(3) Suppose $f(x)$ and $g(x)$ are defined on the same interval $[p,q]$. If the function $F(x)=f(x)-g(x)$ has two distinct zeros in the interval, then $f(x)$ and $g(x)$ are said to be "$\Omega$ functions" on $[p,q]$. If $f(x)$ and $g(x)$ are "$\Omega$ functions" on $[0,3]$, find the range of $m$.

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