
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 one focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ coincides with the focus of the parabola $y^{2}=4 \sqrt {3}x$, and the point $(\sqrt {3}, \frac{1}{2})$ is on the ellipse $C$.
(1) Find the equation of the ellipse $C$;
(2) Let line $l$: $y=kx+m(k\neq 0)$ intersect the ellipse at points $P$ and $Q$. A rhombus with $PQ$ as one of its diagonals has one vertex at $(-1,0)$. If $|PQ|= \frac{2 \sqrt {6}}{3}$, find the value of $k$.

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