
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 $\overrightarrow{a}=(m,-1)$ and $\overrightarrow{b}=( \frac{1}{2}, \frac{ \sqrt{3}}{2})$,
(1) If $m=-\sqrt{3}$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$;
(2) Assume $\overrightarrow{a} \perp \overrightarrow{b}$.
$\quad$(a) Find the value of the real number $m$;
$\quad$(b) If there exist non-zero real numbers $k$ and $t$ such that $[ \overrightarrow{a}+(t^{2}-3) \overrightarrow{b}] \perp (-k \overrightarrow{a}+t \overrightarrow{b})$, find the minimum value of $\frac{k+t^{2}}{t}$.

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