
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 coordinates of point $M(x,y)$ satisfy the following system of inequalities:
$$
\begin{cases}
2x + y - 4 \geqslant 0, \\
x - y - 2 \leqslant 0, \\
y - 3 \leqslant 0,
\end{cases}
$$
and $N$ is any point on the line $y = -2x + 2$, determine the minimum value of the distance $|MN|$.

A) $\frac{\sqrt{5}}{5}$
B) $\frac{2\sqrt{5}}{5}$
C) $1$
D) $\frac{\sqrt{17}}{2}$

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