
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
Define: The sum of the coefficients of the quadratic terms is $1$, the symmetric axes are the same, and the intersection points with the $y$-axis are also the same for two quadratic functions, then they are friendly coaxial quadratic functions. For example, the friendly coaxial quadratic function of $y=2x^{2}+4x-5$ is $y=-x^{2}-2x-5$.<br/>$(1)$ The symmetric axis of the function $y=-3x^{2}+3x+1$ is ______, its friendly coaxial quadratic function is ______, and the relationship between the coefficients of the quadratic terms of the two functions is ______;<br/>$(2)$ Given a quadratic function $C_{1}y=ax^{2}+4ax+4$ (where $a\neq 0$, $a\neq 1$, and $a\neq \frac{1}{2}$), its friendly coaxial quadratic function is denoted as $C_{2}$.<br/>① If the graphs of functions $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$ (the abscissa of point $A$ is less than that of point $B$), find the length of segment $AB$;<br/>② When $-3\leqslant x\leqslant 0$, the difference between the maximum and minimum values of function $C_{2}$ is $8$, find the value of $a$.

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