
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 an ellipse $C$ with foci $F_{1}(-1,0)$, $F_{2}(1,0)$, and passing through the point $E(1, \frac {3}{2})$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ A line $l$ passing through $F_{1}$ intersects the ellipse $C$ at points $A$, $B$ (with point $A$ above the $x$-axis), if $\overrightarrow{AF_{1}}=\lambda \overrightarrow{F_{1}B}$, and $\frac {5}{3}\leqslant \lambda\leqslant \frac {7}{3}$, find the range of values for the slope $k$ of line $l$.

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