
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
An ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ has its left focus $F_{1}$ and right focus $F_{2}$. The eccentricity $e= \frac{1}{2}$. $P$ is a point on the ellipse, and the incircle of $\triangle PF_{1}F_{2}$ has a radius of $1$ and center $I$, with $S_{\triangle PIF_{1}}+S_{\triangle PIF_{2}}=2$.

1. Find the equation of ellipse $E$.
2. Two chords $AB$ and $CD$ pass through the left focus $F_{1}$ of the ellipse, and they are perpendicular to each other. Find the minimum value of $| \overrightarrow{AB}|+| \overrightarrow{CD}|$.

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