
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
For any irrational number that lies between two integers, we define: if the irrational number $T: m \lt T \lt n$ (where $m$ is the largest integer satisfying the inequality, and $n$ is the smallest integer satisfying the inequality), then the "Ya interval" of the irrational number $T$ is defined as $\left(m,n\right)$. For example, $1 \lt \sqrt{2} \lt 2$, so the "Ya interval" of $\sqrt{2}$ is $\left(1,2\right)$.

$(1)$ The "Ya interval" of the irrational number $-\sqrt{7}$ is ______;
$(2)$ If the "Ya interval" of a certain irrational number is $\left(m,n\right)$, and it satisfies $0 \lt m+\sqrt{n} \lt 12$, where $\left\{{\begin{array}{l}{x=m}\\{y=\sqrt{n}}\end{array}}\right.$ is a set of positive integer solutions to the linear equation about $x$ and $y$, $mx-ny=c$, then the value of $c$ is ______.

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