
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 $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(-2,m)$, $\overrightarrow{c}=\overrightarrow{a}+(t^{2}+1)\overrightarrow{b}$, $\overrightarrow{d}=-k\overrightarrow{a}+\frac{1}{t}\overrightarrow{b}$, where $m\in\mathbb{R}$, and $k,t$ are positive real numbers.

1. If $\overrightarrow{a}//\overrightarrow{b}$, find the value of $m$.
2. If $\overrightarrow{a}\perp\overrightarrow{b}$, find the value of $m$.
3. When $m=1$, if $\overrightarrow{x}\perp\overrightarrow{y}$, find the minimum value of $k$.

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