
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 point $M(x\_0, y\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.

(1) Find the trajectory equation of point $P(x,y)$;

(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0$.

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