
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 a moving circle $C$ passes through the point $F(1,0)$ and is tangent to the line $x=-1$.
- (I) Find the equation of the trajectory of the center $C$ of the moving circle; and find the equation of the circle $C_{1}$ when the area of circle $C$ is minimized;
- (II) Let the trajectory curve of the center $C$ of the moving circle be $E$, and the line $y= \frac {1}{2}x+b$ intersects with circle $C_{1}$ and curve $E$ at four different points, from left to right, denoted as $A$, $B$, $C$, $D$, where $B$, $D$ are the intersection points of the line with curve $E$. If the angles of inclination of lines $BF$, $DF$ are complementary, find the value of $|AB|+|CD|$.

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