
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]


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Suppose a quadratic function $y=f(x)$ attains its minimum value $\frac{-t^2}{4}$ at $x = \frac{t+2}{2}$ (where $t \neq 0$), and $f(1) = 0$.

(1) Determine the expression for $y=f(x)$.

(2) If the function $y=f(x)$ takes on its minimum value of $-5$ over the interval $\left[-1, \frac{1}{2}\right]$, find the value of $t$.

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