
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
Let ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ have an eccentricity of $e = \frac{\sqrt{2}}{2}$. Point $A$ is a point on the ellipse, and the sum of the distances from point $A$ to the two foci of ellipse $C$ is $4$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ A moving point $P(x_0,y_0)$ on ellipse $C$ has its symmetric point with respect to the line $y=2x$ as $P_1(x_1,y_1)$. Find the range of values for $3x_1-4y_1$.

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