
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, let the quadratic function be $y=ax^{2}-(2a-2)x-3a-1$ (where $a$ is a constant, and $a \gt 0$). <br/>$(1)$ If the graph of the quadratic function intersects the line $y=-3a-2$ at exactly one point, find the value of the algebraic expression $a^2+\frac{1}{a^2}$; <br/>$(2)$ If points $M(m,y_{1})$ and $N(n,y_{2})$ are two distinct points on the graph of the quadratic function, and $m+n=-2$, and $T=y_{1}+y_{2}$, find the range of values for $T$.

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