
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 equations of curve $C_1$ as $\begin{cases} x = 1 + \frac{1}{2}t \\ y = \frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter) and the polar coordinate equation of curve $C_2$ as $\rho^2 = \frac{12}{3 + \sin^2 \theta}$, establish a polar coordinate system with the origin $O$ of the rectangular coordinate system as the pole and the positive half of the $x$-axis as the polar axis.

(I) Find the Cartesian equation of curve $C_1$ and the polar coordinate equation of curve $C_2$.

(II) If curves $C_1$ and $C_2$ intersect at points $A$ and $B$, and point $F(1, 0)$, find the value of $\frac{1}{|FA|} + \frac{1}{|FB|}$.

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