
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 that the square root of $x-1$ is $\pm 2$, the cube root of $4x+y$ is $3$, and the arithmetic square root of $y-x$ is $m$.  
$(1)$ Find the value of $m$;  
$(2)$ If $5+m=a+b$, where $a$ is an integer and $0 \lt b \lt 1$, find $a-({\sqrt{2}-b})^2$.

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