
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
Given that $x$ is a real number, let $\lfloor x \rfloor$ denote the largest integer not greater than $x$. For example, $\lfloor 1.2 \rfloor = 1$, $\lfloor -1.2 \rfloor = -2$, $\lfloor 1 \rfloor = 1$. If there exists a real number $m \in R$ and $m \notin Z$ such that $f(m) = f(\lfloor m \rfloor)$, then the function $f(x)$ is called an $\Omega$ function.

(I) Determine whether the functions $f(x) = x^2 - \frac{1}{3}x$ and $g(x) = \sin(\pi x)$ are $\Omega$ functions. (Only the conclusion needs to be written)

(II) Given $f(x) = x + \frac{a}{x}$, find a value of $a$ such that $f(x)$ is an $\Omega$ function, and provide a proof.

(III) Suppose function $f(x)$ is defined on $R$ and is periodic with the smallest positive period $T$. If $f(x)$ is not an $\Omega$ function, find the minimum value of $T$.

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