
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
If the sum of the solutions of two linear equations is $1$, we call these two equations "beautiful equations". For example, the equations $2x-1=3$ and $x+1=0$ are "beautiful equations".  
$(1)$ Are the equations $4x-(x+5)=1$ and $-2y-y=3$ "beautiful equations"? Please explain the reason.  
$(2)$ If the equations $2x-n+3=0$ and $x+5n-1=0$ regarding $x$ are "beautiful equations", find the value of $n$.

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