
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 line $l: x + \sqrt{2}y = 4\sqrt{2}$ and an ellipse $C: mx^2+ny^2=1$ ($n > m > 0$) that intersect at exactly one point $M[2\sqrt{2}, 2]$.  
(1) Find the equation of the ellipse $C$.  
(2) Let $A$ and $B$ be the left and right vertices of the ellipse $C$ respectively, and $O$ be the coordinate origin. A moving point $Q$ is such that $QB \perp AB$. The line $AQ$ intersects the ellipse at point $P$. Find the value of $\vec{OQ} \cdot \vec{OP}$.

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