
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: $T(x,y) = axy + bx - 4$ (where $a$ and $b$ are non-zero constants). For example, $T(0,1) = a \times 0 \times 1 + b \times 0 - 4 = -4$. If $T(2,1) = 2$ and $T(-1,2) = -8$, which of the following conclusions are correct?

① $a=1$, $b=2$;

② If $T(m,n) = 0$ (where $n \neq -2$), then $m = \frac{4}{n+2}$;

③ If $T(m,n) = 0$, then $m$ and $n$ have exactly $3$ sets of integer solutions;

④ If the value of $T(kx,y)$ remains unchanged regardless of the value of $k$, then $y = -2$;

⑤ If $T(kx,y) = T(ky,x)$ holds for any rational numbers $x$ and $y$, then $k=0$.

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