
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 $f(x)=ax^{2}+bx+3$ that attains its minimum value at $x=2$, and the length of the line segment intercepted by the function's graph on the $x$-axis is $2$.
(1) Find the analytical expression for the function $f(x)$;
(2) If one zero of the function $g(x)=f(x)-mx$ lies in the interval $(0,2)$, and the other zero lies in the interval $(2,3)$, find the range of real number $m$;
(3) When $x \in [t,t+1]$, the minimum value of the function $f(x)$ is $-\frac{1}{2}$. Find the value of real number $t$.

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