
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 with its foci on the x-axis and an eccentricity of $\frac{\sqrt{3}}{2}$, the distance from the right focus to the right directrix ($x= \frac{a^{2}}{c}$) is $\frac{\sqrt{3}}{3}$.  
(1) Find the standard equation of the ellipse;  
(2) A line passes through one focus of the ellipse and has a slope of 1, find the distance between the two intersection points of the line and the ellipse.

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