
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, the graph of a linear function $y=kx+b\left(k\neq 0\right)$ intersects the $x$-axis and $y$-axis at points $A\left(-3,0\right)$ and $B\left(0,-3\right)$, respectively. The graph of a quadratic function $y=x^{2}+mx+n$ passes through point $A$.  
$(1)$ Find the expression of the linear function $y=kx+b$;  
$(2)$ If the graph of the quadratic function $y=x^{2}+mx+n$ intersects the $y$-axis at point $\left(0,3\right)$, determine whether the vertex of this quadratic function lies on the graph of the line $y=kx+b\left(k\neq 0\right)$;  
$(3)$ When $n \gt 0$ and $m\leqslant 5$, the minimum value of the quadratic function $y=x^{2}+mx+n$ is $t$. Find the range of values for $t$.

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