
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 line passing through the point $\left(0,5\right)$ and parallel to the $x$-axis intersects the line $y=x$ at point $P$. The point $P$ is symmetric with respect to the line $x=2$ to point $Q$. The parabola $y=ax^{2}+bx$ passes through points $P$ and $Q$.  
$(1)$ The coordinates of point $P$ are ______; the coordinates of point $Q$ are ______.  
$(2)$ Find the expression of the parabola $y=ax^{2}+bx$.  
$(3)$ If point $A$ lies on the parabola $y=ax^{2}+bx$ and has an $x$-coordinate of $2m$, a perpendicular line is drawn from point $A$ to the line $x=2$, with the foot of the perpendicular being $B$. When points $A$ and $B$ do not coincide, a rectangle $ABCD$ is constructed with $AB$ as a side such that $BC=4AB$.  
① When the center of rectangle $ABCD$ lies exactly on the parabola $y=ax^{2}+bx$, find the value of $m$.  
② When the parabola $y=ax^{2}+bx$ intersects $BC$, let the intersection point be $E$. If $cos∠BAE=\frac{{\sqrt{5}}}{5}$, directly write down the value of $m$.

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