
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
Given matrices $$A= \begin{bmatrix} x & y \\ 1 & 2\end{bmatrix}$$ and $$B= \begin{bmatrix} -1 & m \\ -2 & m\end{bmatrix}$$, and vector $$\alpha= \begin{bmatrix} 2 \\ 3\end{bmatrix}$$, where $x, y \in (0, +\infty)$, if $A\alpha=B\alpha$, find the maximum value of $xy$.

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