
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 the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).

1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$.

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