
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 the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), where \(a > b > 0\) and \(c^{2}+b^{2}=a^{2}\), a line passing through the focus intersects the ellipse at points \(M\) and \(N\). The length of line segment \(\mathrm{MN}\) is within the range \([3,4]\).

(1) Find the equation of the ellipse.
(2) Let \(\mathrm{D}\) and \(\mathrm{E}\) be two points on the ellipse that are symmetric with respect to the \(\mathrm{x}\)-axis. For any two points \(A\left(x_{1}, 0 \right)\) and \(B\left(x_{2}, 0 \right)\) on the \(\mathrm{x}\)-axis, such that \(D A\) does not tangent to the ellipse, the intersection point \(C\) of lines \(D A\) and \(B E\) still lies on the ellipse. Find the relationship between \(x_{1}\) and \(x_{2}\).

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