
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 polar coordinate system, the pole is the origin of the Cartesian coordinate system $xOy$, and the polar axis is the positive half-axis of $x$. The unit of length is the same in both coordinate systems. It is known that the polar equation of curve $C$ is $\rho=2(\cos \theta+\sin \theta)$.

$(1)$ Find the Cartesian coordinate equation of $C$;

$(2)$ Line $l$: $\begin{cases} x= \frac{1}{2}t \\ y=1+ \frac{ \sqrt{3}}{2}t \end{cases}$ (where $t$ is a parameter) intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $E$. Find $|EA|+|EB|$.

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