
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 that the eccentricity of the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\left(a > b > 0\right)$ is $e= \frac{\sqrt{2}}{2}$, and point $A$ is a point on the ellipse such that the sum of the distances from point $A$ to the two foci of ellipse $C$ is $4$.

(1) Find the equation of the ellipse $C$;

(2) If a point $P(x_{0},y_{0})$ on the ellipse $C$ is symmetric about the line $y=2x$ to the point $P_{1}(x_{1},y_{1})$, find the range of $3x_{1}-4y_{1}$.

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