
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 $m$ and $n$ respectively represent the numbers obtained when a fair cubic die (with faces numbered $1$, $2$, $3$, $4$, $5$, $6$) is rolled twice in succession, with $m$ being the result of the first roll and $n$ the result of the second.  
$(1)$ Find the probability that the equation $x^{2}+mx+n^{2}=0$ has two distinct real roots;  
$(2)$ Find the probability that the ratio $\frac{m}{n}$ is not an integer.

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