
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 $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $x-3$, and $y_{2}$ is directly proportional to $x^{2}+1$. When $x=0$, $y=-2$, and when $x=1$, $y=4$.  
$(1)$ Find the functional relationship between $y$ and $x.  
$(2)$ Find the coordinates of the intersection points of the function with the coordinate axes.

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