
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 the quadratic equation in $x$: $x^{2}-4x+2k=0$.  
$(1)$ If the equation has real roots, find the range of real values for $k$.  
$(2)$ If $k$ is the largest integer that satisfies $(1)$, and one of the roots of the equation $x^{2}-4x+2k=0$ is a root of the quadratic equation $x^{2}-2mx+3m-1=0$, find the value of $m$ and the other root of the equation.

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