
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 a focus intersects the ellipse at points \(M\) and \(N\), and the length of segment \(\mathrm{MN}\) ranges from 3 to 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 \(x\)-axis. For any two points \(A\left(x_{1}, 0\right)\) and \(B\left(x_{2}, 0\right)\) on the \(x\)-axis, when \(\mathrm{DA}\) does not touch the ellipse, the intersection of lines \(\mathrm{DA}\) and \(\mathrm{BE}\) still lies on the ellipse. Find the relationship between \(x_{1}\) and \(x_{2}\).

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