
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
For the one-variable quadratic equation $x^{2}-3x+k=0$ with real roots:<br/>$(1)$ Find the range of values for $k$;<br/>$(2)$ If $k$ is the largest integer that satisfies the condition, and the one-variable quadratic equation $\left(m-1\right)x^{2}+x+m-3=0$ shares a root with the equation $x^{2}-3x+k=0$, find the value of $m$ in this case.

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