
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 two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in a plane that are orthogonal to each other, with magnitudes $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=1$, and $k$ and $t$ are real numbers that are not simultaneously zero.
(1) If the vectors $\overrightarrow{x} = \overrightarrow{a} + (t-3)\overrightarrow{b}$ and $\overrightarrow{y} = -k\overrightarrow{a} + t\overrightarrow{b}$ are orthogonal, find the functional relationship $k = f(t)$;
(2) Find the minimum value of the function $k = f(t)$.

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