
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 $a$ and $b$ in a plane that are orthogonal to each other, with $|a|=2$ and $|b|=1$. Let $k$ and $t$ be real numbers that are not simultaneously zero.

(1) If $x=a+(t-3)b$ and $y=-ka+tb$ 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}
user