
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
We define: if the graphs of two linear functions both pass through the same point on the coordinate axes, then we call these two linear functions "intersecting axis linear functions". For example, the graphs of the linear functions $y=2x-3$ and $y=-x-3$ both pass through the same point $(0,-3)$ on the $y$-axis, so these two functions are "intersecting axis linear functions". Another example is the linear functions $y=-x-2$ and $y=3x+6$ whose graphs both pass through the same point $(-2,0)$ on the $x$-axis, so these two functions are "intersecting axis linear functions".

$(1)$ Are the linear functions $y=3x+1$ and $y=3x-1$ "intersecting axis linear functions"? If yes, please explain the reason; if not, also explain the reason and write down one of the functions as an "intersecting axis linear function".

$(2)$ Given the linear functions $y_{1}=-3x+3$, $y_{2}=4x+b$, if $y_{1}$ and $y_{1}-y_{2}$ are "intersecting axis linear functions", find the value of $b$.

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