
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 $y=\frac{k}{x}$ be a function where $k$ is a constant and $k\neq 0$. Point $M(3,a)$ lies on the graph of this function. Point $M$ is first reflected across the $y$-axis and then translated down by $4$ units to obtain point $N$, which also lies on the graph of the function. <br/>$(1)$ Find the expression for this function;<br/>$(2)$ If $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$ are the coordinates of three points on the graph of the function obtained in part $(1)$, and satisfy $x_{3} \gt x_{2} \gt x_{1} \gt 0$, compare $y_{1}+y_{2}$ with $2y_{3}$ and provide a reason for your comparison.

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