
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 eccentricity $e = \frac{\sqrt{2}}{2}$. Point A is a point on the ellipse, and the sum of distances from point A to the two foci of ellipse C is 4.
(1) Find the equation of ellipse C.
(2) If P$(x_0, y_0)$ is a moving point on the ellipse C and its symmetric point with respect to the line $y = 2x$ is P$_1(x_1, y_1)$, find the range of values for $3x_1 - 4y_1.$

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