
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 vectors $a=(m,1)$ and $b=(\frac{1}{2}, \frac{\sqrt{3}}{2})$,
(1) Find the value of the real number $m$ if vector $a$ is parallel to vector $b$;
(2) Find the value of the real number $m$ if vector $a$ is perpendicular to vector $b$;
(3) If $a \perp b$, and there exist non-zero real numbers $k$ and $t$ such that $[a+(t^2)(-3)b] \perp (-ka+tb)$, find the minimum value of $\frac{k+t^2}{t}$.

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