
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 two equations about $x$: $3\left(2x-1\right)=k+2x$ and $\frac{x-k}{2}=x+2k$.<br/>$(1)$ If the solution to the equation $3\left(2x-1\right)=k+2x$ is $x=4$, find the solution to the equation $\frac{x-k}{2}=x+2k$;<br/>$(2)$ If the solutions to the equations $3\left(2x-1\right)=k+2x$ and $\frac{x-k}{2}=x+2k$ are the same, find the value of $k$.

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