
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
Let $m\in \mathbb{R}$, given the line $l_{1}$: $\left(m+1\right)x+my+2-m=0$, a line $l_{2}$ is drawn passing through the point $\left(1,2\right)$, and $l_{1}$ <span class="special-symbol">∥</span> $l_{2}$. Find the maximum value of the distance between the lines $l_{1}$ and $l_{2}$.

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