
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 circle $M$: $x^2 + y^2 + 2\sqrt{2}y - 10 = 0$ and point $N(0, \sqrt{2})$, $Q$ is any point on circle $M$, and the perpendicular bisector of line segment $NQ$ intersects with $QM$ at point $P$. The locus of $P$ forms curve $E$.

(1) Find the equation of curve $E$;

(2) Point $A$ is the intersection point of curve $E$ and the positive semi-axis of $x$. Line $x = ty + m$ intersects $E$ at points $B$ and $C$. The slopes of lines $AB$ and $AC$ are $k_1$ and $k_2$, respectively. If $k_1 \cdot k_2 = 9$, find: $\quad\quad$ (i) the value of $m$; $\quad\quad$ (ii) the maximum area of $\Delta ABC$.

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