
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
Let $m$ and $n$ be real numbers. Given that ${z_1}=1+\sqrt{2}i$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}+mx+n=0$. <br/>$(1)$ Find the values of $m$ and $n$;<br/>$(2)$ Let $z_{2}$ be the other root of the equation, and let the vectors corresponding to the complex numbers $z_{1}$ and $z_{2}$ be $\overrightarrow{a}$ and $\overrightarrow{b}$, respectively. If the angle between the vectors $t\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+t\overrightarrow{b}$ is acute, find the range of the real number $t$.

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