
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 that point $A(1,1)$ is a point on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\_1$, $F\_2$ are the two foci of the ellipse such that $|AF\_1|+|AF\_2|=4$.
(I) Find the standard equation of the ellipse;
(II) Find the equation of the tangent line to the ellipse that passes through $A(1,1)$;
(III) Let points $C$ and $D$ be two points on the ellipse such that the slopes of lines $AC$ and $AD$ are complementary. Determine whether the slope of line $CD$ is a constant value. If it is, find the value; if not, explain the reason.

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