
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 $m\in R$, the moving straight lines $({l}_{1}:x+my-1=0)$ and $({l}_{2}:mx-y-2m+3=0)$ intersect at point $P$, and pass through points $A$ and $B$ respectively. Determine the maximum value of $|PA|+|PB|$.

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