
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
Definition: Two quadratic coefficients sum to $1$, have the same axis of symmetry, and intersect the $y$-axis at the same point, then the quadratic functions are friendly coaxial quadratic functions. For example, the friendly coaxial quadratic function of $y=2x^{2}+4x-5$ is $y=-x^{2}-2x-5$.  
$(1)$ The friendly coaxial quadratic function of the function $y=\frac{1}{4}{x^2}-2x+3$ is ______.  
$(2)$ When $-1\leqslant x\leqslant 4$, the friendly coaxial quadratic function of the function $y=\left(1-a\right)x^{2}-2\left(1-a\right)x+3(a\neq 0$ and $a\neq 1)$ has a maximum value of $5$. Find the value of $a$.  
$(3)$ Given points $\left(m,p\right)$ and $\left(m,q\right)$ lie on the graph of the quadratic function ${y_1}=a{x^2}+4ax+c(a>\frac{1}{2}$ and $a\neq 1)$ and its friendly coaxial quadratic function $y_{2}$, compare the values of $p$ and $q$, and explain the reason.

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