
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 $x$ be a real number, and use $\left[x\right]$ to represent the largest integer not exceeding $x$. <br/>$(1)$ If the function $f\left(x\right)=\left[x\right]$, find the values of $f\left(1.2\right)$ and $f\left(-1.2)$; <br/>$(2)$ If the function $f(x)=[{\frac{{x+1}}{2}}]-[{\frac{x}{2}}]$ ($x\in R$), find the range of $f\left(x\right)$; <br/>$(3)$ If there exists $m\in R$ and $m\notin Z$ such that $f\left(m\right)=f\left(\left[m\right]\right)$, then the function $f\left(x\right)$ is called an $\Omega$ function. If the function $f(x)=x+\frac{a}{x}$ is an $\Omega$ function, find the range of values for $a$.

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