
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
Read the following content: We have learned "Quadratic Radical" and "Multiplication Formula". Smart you can find: when $a \gt 0$, $b \gt 0$, $\because {({\sqrt{a}-\sqrt{b}})^2}=a-2\sqrt{ab}+b≥0$, $\therefore a+b≥2\sqrt{ab}$, and the equality holds only when $a=b$. For example, when $a \gt 0$, find the minimum value of $a+\frac{4}{a}$. Solution: $\because a \gt 0$, $\therefore a+\frac{4}{a}≥2\sqrt{a⋅\frac{4}{a}}$. Also, $\because 2\sqrt{a⋅\frac{4}{a}}=4$, $\therefore a+\frac{4}{a}≥4$, with equality when $a=2$. Therefore, the minimum value of $a+\frac{4}{a}$ is $4$. Please use the above conclusion to solve the following problems: <br/>$(1)$ When $x \gt 0$, the value of $x$ that makes $x+\frac{1}{x}$ have a minimum value of $2$ is $x=$______. <br/>$(2)$ When $m \gt 0$, find the minimum value of $\frac{{{m^2}+5m+12}}{m}$.

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