
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
Definition: An equation of the form $ax+y=b$ (where $a$ and $b$ are constants, and $x$ and $y$ are variables) is called a "beautiful two-variable linear equation." When $y=2x$, the value of $x$ in the "beautiful two-variable linear equation $ax+y=b$" is called the "beautiful value" of the equation. For example, when $y=2x$, the "beautiful two-variable linear equation" $3x-y=4$ becomes $3x-2x=4$, which gives $x=4$, so its "beautiful value" is $4$.

$(1)$ Find the "beautiful value" of the "beautiful two-variable linear equation" $5x-y=1$;  
$(2)$ If the "beautiful value" of the "beautiful two-variable linear equation" $\frac{1}{3}x+y=m$ is $-3$, find the value of $m$;  
$(3)$ Does there exist an $n$ such that the "beautiful value" of the beautiful two-variable linear equation $\frac{5}{2}x+y=n$ is the same as the "beautiful value" of the beautiful two-variable linear equation $4x-y=n-2$? If it exists, find the value of $n$ and the "beautiful value" at that time; if it does not exist, explain the reason.

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