
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
Define: If $x$ and $y$ satisfy $x^{2}=4y+t$ and $y^{2}=4x+t$ with $x\neq y$ (where $t$ is a constant), then the point $M\left(x,y\right)$ is called a "harmonic point." <br/>$(1)$ If $P\left(3,m\right)$ is a "harmonic point," then $m=$______; <br/>$(2)$ If the hyperbola $y=\frac{k}{x}(-3 \lt x \lt -1)$ has a "harmonic point," then the range of values for $k$ is ______.

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