
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with its right focus at $F(1,0)$, and the endpoints of the minor axis are $B_{1}$, $B_{2}$, and $\overrightarrow{FB_{1}} \cdot \overrightarrow{FB_{2}}=-a$.
1. Find the equation of the ellipse $C$;
2. A line $l$ passing through point $F$ with slope $k(k \neq 0)$ intersects the ellipse at points $M$ and $N$. The perpendicular bisector of chord $MN$ intersects the $x$-axis at point $D$. Let $P$ be the midpoint of chord $MN$, find the range of values for $\frac{|DP|}{|MN|}$.

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