
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 the equation $x^{2}+y^{2}-2x+4y+4m=0$.  
$(1)$ If this equation represents a circle, find the range of real numbers for $m$.  
$(2)$ If the value of $m$ is the largest integer that can be taken in $(1)$, then the obtained circle is denoted as circle $E$. If circle $E$ is symmetric about the $y$-axis with circle $F$, and $P\left(x,y\right)$ is any point on circle $F$, find the maximum and minimum distance from $P\left(x,y\right)$ to the line $x+y-1=0$.

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