
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
Given the inverse proportion function $y=\frac{k-4}{x}$ passes through the first and third quadrants.<br/>$(1)$ Find the range of $k$;<br/>$(2)$ If $a \gt 0$, the graph of this function passes through two points in the first quadrant $(a+5$,$y_{1})(2a+1$,$y_{2})$, and $y_{1} \lt y_{2}$, find the range of $a$.

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