
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 the solution set of the inequality $|x+1| \gt a(a\in N^{*})$ be $A$, and $\frac{3}{2}∈A$, $\frac{1}{2}∉A$.  
$(1)$ Find the value of $a$;  
$(2)$ If $m$, $n$, $s$ are positive real numbers, and $m+n+\sqrt{2}s=a$, find the minimum value of $m^{2}+n^{2}+s^{2}$.

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