
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{5}}{5}$, and the equation of the right directrix is $x=5$.

(1) Find the equation of the ellipse;  
(2) A line $l$ with a slope of 1 passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse at points $A$ and $B$. Let $P$ be a moving point on the ellipse. Find the maximum area of $\triangle PAB$.

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