
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]


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Given the function $y=\left[x\right]$, which is called the Gaussian function and represents the greatest integer not exceeding $x$, such as $\left[\pi \right]=3$, $\left[-2.5\right]=-3$. The solution set of the inequality $\frac{[x]}{[x]-4}＜0$ is ______; when $x \gt 0$, the maximum value of $\frac{[x]}{[x]^2+4}$ is ______.

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