
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
**(1)** Calculate: $\int_{0}^{1}(\sqrt{1-x^{2}}+e^{x})dx=$ _________.

**(2)** The number of five-digit even numbers that can be formed without repetition of digits from $0,1,2,3,$ and $4$ is _________.

**(3)** For a real number $x$, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Observe the following equations:
$\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor = 3$
$\lfloor \sqrt{4} \rfloor + \lfloor \sqrt{5} \rfloor + \lfloor \sqrt{6} \rfloor + \lfloor \sqrt{7} \rfloor + \lfloor \sqrt{8} \rfloor = 10$
$\lfloor \sqrt{9} \rfloor + \lfloor \sqrt{10} \rfloor + \lfloor \sqrt{11} \rfloor + \lfloor \sqrt{12} \rfloor + \lfloor \sqrt{13} \rfloor + \lfloor \sqrt{14} \rfloor + \lfloor \sqrt{15} \rfloor = 21$
...

Following this pattern, the equation for the $n$-th term is __________.

**(4)** Let the function $f(x)=x^{3}-2ex^{2}+mx-\ln x$, and let $g(x)=\frac{f(x)}{x}$. If the function $g(x)$ has at least one zero, then the range of the real number $m$ is __________.

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