
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 a polar coordinate system with its pole at the origin $O$ of the rectangular coordinate system, the positive semi-axis of the $x$-axis as the polar axis, and both coordinate systems having the same unit length, the parametric equations of line $l$ are $\begin{cases} & x=\frac{1}{2}+t\cos \alpha \\ & y=t\sin \alpha \\ \end{cases} (t \text{ is a parameter}, 0 < \alpha < \pi )$, and the polar equation of curve $C$ is $\rho \sin^{2}\theta =2\cos \theta$.

(1) Find the rectangular coordinate equation of curve $C$.

(2) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, and there is a fixed point $P(\frac{1}{2},0)$. When $\alpha = \frac{\pi}{3}$, find the value of $|PA|+|PB|$.

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