
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, a parabola $y=x^{2}+bx+c$ (where $b$ and $c$ are constants) passes through the points $(3,0)$ and $(0,-3)$.  
$(1)$ Find the expression of the parabola function.  
$(2)$ When $-1\leqslant x\leqslant 4$, find the maximum and minimum values of the quadratic function $y=x^{2}+bx+c$.  
$(3)$ Let point $P$ be any point on the graph of this function with an abscissa of $m$. Draw a line segment $PQ$ perpendicular to the $y$-axis passing through point $P$, intersecting the line $x=3$ at point $Q$. When points $P$ and $Q$ are not coincident, construct an isosceles right triangle $PQM$ with $PQ$ as the base and point $P$ as the right angle vertex, extending towards the negative $y$-axis.  
① When the distance from point $M$ to the line of the ordinate of the parabola vertex is $5$, find the value of $m$.  
② When the maximum difference in the ordinate of the points inside the isosceles right triangle $PQM$ is $1$, directly write down the value of $m$.

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