
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 a new operation $T$ for $x$ and $y$ as follows: $T(x,y)=(mx+ny)(x+2y)$ (where $m$ and $n$ are non-zero constants). For example, $T(1,1)=3m+3n$. Given that $T(1,-1)=0$ and $T(0,2)=8$:
1. Find the values of $m$ and $n$.
2. If the system of inequalities in terms of $p$ is $\left\{\begin{array}{l}T(2p,\;\;2-p)>4\\ T(4p,\;\;3-2p)\leq a\end{array}\right.$ has exactly $3$ integer solutions, find the range of values for $a$.
3. When $x^{2}\neq y^{2}$, $T(x,y)=T(y,x)$ holds for any rational numbers $x$ and $y$. Write down the relationship that $m$ and $n$ satisfy directly.

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