
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 points $A(-3,5)$ and $B(2,15)$, $P$ is a moving point on the line $x-y+1=0$.  
$(1)$ Find the symmetric point ${A'}(x_{0}, y_{0})$ of point $A(-3,5)$ with respect to the line $x-y+1=0$;  
$(2)$ Find the minimum value of $|PA|+|PB|$.

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