
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 line $l$ as $\begin{cases} x=\frac{1}{2}t \\ y=1+\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and the polar equation of curve $C$ as $\rho =2\sqrt{2}\sin \left( \theta +\frac{\pi }{4} \right)$, line $l$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $P$.

(1) Find the Cartesian equation of curve $C$.

(2) Calculate the value of $\frac{1}{\left| PA \right|}+\frac{1}{\left| PB \right|}$.

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