
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 ${C}_{1}:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$ and a hyperbola ${C}_{2}:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$, a tangent line $l$ is drawn passing through a point $P(1,\frac{3}{2})$ on the ellipse $C_{1}$. The line $l$ intersects the $x$-axis at point $A$ and intersects the two asymptotes of the hyperbola $C_{2}$ at points $D$ and $E$, where $D$ is the midpoint of $AE$. Find the eccentricity of the hyperbola $C_{2}$.

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