
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 a quadratic function $y=\frac{1}{2}(m-1)x^{2}+\left(n-6\right)x+1$ $(m\geqslant 0, n\geqslant 0)$, determine the maximum value of $mn$ given that $y$ decreases as $x$ increases when $1\leqslant x\leqslant 2$.

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