
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
For any non-empty set $A$, define: $\Omega \left(A\right)=\{\pi \left(X\right)\left|\right.X\subseteq A,X\neq \varnothing \}$, where $\pi \left(X\right)$ represents the product of all elements in the non-empty set $X$, especially, if $X=\{x\}$, then $\pi \left(X\right)=x$.  
$(1)$ If $A_{1}=\{\frac{1}{2}$, $1, 4\}$, $A_{2}=\{2, 3, 5\}$, please directly write down the number of elements in the sets $\Omega (A_{1})$ and $\Omega (A_{2})$;  
$(2)$ If $A=\{a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}\}$, where $a_{i}$ is a positive integer $\left(i=1,2,3,4,5\right)$, find the maximum and minimum number of elements in the set $\Omega \left(A\right)$, and explain the reasons;  
$(3)$ If $A=\{a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$, $a_{7}\}$, where $a_{i}$ is a positive real number $\left(i=1,2,3,4,5,6,7\right)$, find the minimum number of elements in the set $\Omega \left(A\right)$, and explain the reasons.

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