
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
If the solutions of two equations differ by $k$, and $k$ is a positive integer, then the equation with the larger solution is called the "$k$-shifted equation" of the other equation. For example, the solution of the equation $x-3=0$ is $x=3$, and the solution of the equation $x-1=0$ is $x=1$. Therefore, the equation $x-3=0$ is the "$2$-shifted equation" of the equation $x-1=0$.

$(1)$ Determine whether the equation $2x-3=0$ is the $k$-shifted equation of the equation $2x-1=0$ ______ (Fill in "yes" or "no");<br/>$(2)$ If the equation $2x+m+n=0$ is the "$2$-shifted equation" of the equation $2x+m=0$, find the value of $n$;<br/>$(3)$ If the equation $5x+b=1$ is the "$3$-shifted equation" of the equation $5x+c=1$, find the value of $2b-2\left(c+3\right)$.

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