
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 that the line $y=kx+b$ can be transformed into $kx-y+b=0$, the distance $d$ from point $P(x_{0}, y_{0})$ to the line $kx-y+b=0$ can be calculated using the formula $d=\frac{{|k{x_0}-{y_0}+b|}}{{\sqrt{1+{k^2}}}}$. For example, find the distance from point $P(-2,1)$ to the line $y=x+1$. Find:

1. The distance from point $P(1,2)$ to the line $y=2x-1$.
2. Given that $M$ is a point on the line $y=x+2$, and the distance from $M$ to the line $y=2x-1$ is $2\sqrt{5}$, find the coordinates of $M$.
3. Given that the minimum distance from a point on the line segment $y=kx+4$ ($-1\leq x\leq 2$) to the line $y=x+2$ is $\sqrt{2}$, find the value of $k$.

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