
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 $\left[x\right)$ represent the smallest integer greater than $x$, such as $\left[3\right)=4$, $\left[-1.2\right)=-1$. Which of the following conclusions is correct? (Select all correct conclusions) $①$ $\left[0\right)=0$; $②$ The minimum value of $\left[x\right)-x$ is $0$; $③$ The maximum value of $\left[x\right)-x$ is $0$; $④$ There exists a real number $x$ such that $\left[x\right)-x=0.5$ holds.

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