
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
Define a new operation "$\oplus $": For all $x$, $y\in R$, $x\oplus y=\lg (10^{x}+10^{y})$.  
$(Ⅰ)$ For any real numbers $a$, $b$, $c$, determine the relationship between $\left(a\oplus b\right)-c$ and $\left(a-c\right)\oplus \left(b-c\right)$;  
$(Ⅱ)$ If the set of integers that are solutions to the inequality $\left(x-1\right)^{2} \gt [(a^{2}x^{2})\oplus (a^{2}x^{2})]-\lg 2$ has exactly $3$ elements, find the range of real numbers $a$;  
$(Ⅲ)$ Given functions $f(x)=\lg\{[x+4⊕(x+4)]-\sqrt{2x+3}-\lg 2\}$ and $g\left(x\right)=\left(1\oplus x\right)\oplus \left(-x\right)$, if for any $x_{1}\in R$, there always exists $x_{2}∈[-\frac{3}{2},+∞)$ such that $g(x_{1})=\lg |3m-2|+f(x_{2})$, find the range of real numbers $m$.

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