
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
In the rectangular coordinate system $x0y$, the parametric equations of the curve $C_1$ are given by $\begin{cases} x=a+acos\phi \\ y=asin\phi \end{cases}$ (where $\phi$ is the parameter, and $0.5\pi\leq\phi\leq1.5\pi$, $a>0$). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of the curve $C_2$ is given by $\rho^2=\frac{4}{1+3sin^{2}\theta}$.

1. Find the Cartesian equation of $C_1$ and the polar equation of $C_2$.
2. If the intersection points of $C_1$ and $C$ are $A$ and $B$, and $|AB|=\frac{4\sqrt{2}}{3}$, find $a$.

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