
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
Let $P$ be a moving point on the circle $O$: $x^{2}+y^{2}=16$, point $D$ is the projection of $P$ on the $x$-axis, and $M$ is a point on $PD$ such that $|MD|= \frac {3}{4}|PD|$.

(1) Find the equation of the trajectory $C$ of point $M$ as $P$ moves on the circle.
(2) Find the length of the line segment intercepted by $C$ on the line passing through point $(2,0)$ with a slope of $\frac {3}{4}$.

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