
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
In the Cartesian coordinate system, if the sum of the horizontal and vertical coordinates of point $P$ is zero, then point $P$ is called a "zero-sum point." Given the quadratic function $y=x^{2}+3x+m$.  
$(1)$ When $m=3$, find the "zero-sum point" on the quadratic function $y=x^{2}+3x+m$;  
$(2)$ If the graph of the quadratic function $y=x^{2}+3x+m$ has exactly one "zero-sum point," find the value of $m$.

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