
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passing through the point $P(-1,-1)$, with $c$ as the semi-focal distance of the ellipse, and $c = \sqrt{2}b$. Through point $P$, two perpendicular lines $l_1$, $l_2$ are drawn to intersect the ellipse $C$ at two other points $M$ and $N$.  
(1) Find the equation of ellipse $C$.  
(2) If the slope of line $l_1$ is $-1$, find the area of $\triangle PMN$.  
(3) If the midpoint of segment $MN$ lies on the x-axis, find the equation of line $MN$.

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