
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 $\sum_{t=1}^{k}{{a}_{t}}=a_{1}+a_{2}+\cdots +a_{k}$, $\underset{\stackrel{k}{π}}{t=1}a_{t}=a_{1}\times a_{2}\times \cdots \times a_{k}$. There exists a positive integer $n$, and $n\geqslant 2$. If a set $A=\{a_{1}$,$a_{2}$,$\cdots $,$a_{n}\}$ satisfies $\sum_{t=1}^{n}a_{t}=\underset{\stackrel{k}{π}}{t=1}a_{t}$, then the set $A$ is called a "harmonic set".<br/>$(1)$ Determine whether the set $E=\{1,2\}$ and the set $F=\{-1,0,1\}$ are "harmonic sets";<br/>$(2)$ Given real numbers $x$ and $y$, if the set $\{x,y\}$ is a "harmonic set", does there exist a real number $z$ such that $z^{2}=xy$, and $\{x,y,z\}$ is also a "harmonic set"? If yes, find all real numbers $z$ that satisfy the conditions. If not, explain why;<br/>$(3)$ If a finite set $M$ is a "harmonic set" and all elements in the set $M$ are positive integers, find all possible sets $M$.

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