
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
New definition: We call the parabola $y=ax^{2}+bx+c$ (where $ab\neq 0$) and the parabola $y=bx^{2}+ax+c$ as "related parabolas." For example, the "related parabola" of the parabola $y=2x^{2}+3x+1$ is $y=3x^{2}+2x+1$. Given the parabola $C_{1}$: $y=4ax^{2}+ax+4a-3$ ($a\neq 0$), the "related parabola" is $C_{2}$.  
$(1)$ Write the analytical expression of $C_{2}$ (expressed in terms of $a$) and the coordinates of the vertex.  
$(2)$ If $a \gt 0$, a point $P$ is taken on the $x$-axis. Perpendicular lines are drawn from point $P$ to intersect the parabolas $C_{1}$ and $C_{2}$ at points $M$ and $N$, respectively.  
① When $MN=6a$, find the coordinates of point $P$.  
② When $a-4\leqslant x\leqslant a-2$, the difference between the maximum and minimum values of $C_{2}$ is $2a$. Find the value of $a$.

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