
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 three non-concurrent lines $l\_1$: $4x+y-4=0$, $l\_2$: $mx+y=0$, $l\_3$: $x-my-4=0$.
(1) Find the value of the real number $m$ when these three lines cannot form a triangle.
(2) Find the distance between the two feet of the perpendiculars when $l\_3$ is perpendicular to both $l\_1$ and $l\_2$.

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