
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 the parabola $G:y=mx^{2}-\left(1-4m\right)x+c$ passes through the points $\left(1,a\right)$, $\left(-1,a\right)$, $\left(0,-1\right)$.  
$(1)$ Find the analytical expression of the parabola.  
$(2)$ If a line has exactly one common point with the parabola and is not parallel to the axis of symmetry of the parabola, then the line is tangent to the parabola. This line is called the tangent line of the parabola. Let the tangent line of the parabola $G$ be $l:y=kx+b\left(k\neq 0\right)$.  
① When the tangent line $l$ passes through the point $D\left(0,-3\right)$, find the analytical expression of the tangent line and the coordinates of the corresponding tangent point.  
② Given that point $F$ is a moving point on the $y$-axis, when the tangent line $l$ intersects the lines $y=2$ and $y=-4$ at points $P$ and $Q$ respectively, and $FP^{2}-FQ^{2}=-4$, find the coordinates of point $F$.

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