
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 some functions, the relationship between the values of the function determined by the size of the independent variable can not only be judged by the graph of the function, but also by algebraic methods. This is a typical application of the "combination of number and shape" concept.  
$(1)$ Given a linear function $y=-2x+1$ with two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$ on its graph, where $x_{1} \lt x_{2}$, how can we determine the relationship between $y_{1}$ and $y_{2}$ using algebraic methods? Since points $A$ and $B$ are on the graph of the function, we have $y_{1}=-2x_{1}+1$ and $y_{2}=-2x_{2}+1$. By subtracting $y_{1}$ and $y_{2}$, write out the reasoning process following this approach.  
$(2)$ Given an inverse proportion function $y=\frac{2}{x}$ with two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$ on its graph, where $x_{1} \lt x_{2} \lt 0$, write out the reasoning process for determining the relationship between $y_{1}$ and $y_{2}$ following the approach in $(1)$.

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