
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 origin $O$ of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is given by $p\sin^2\theta=4\cos\theta$;

$(I)$ Find the Cartesian equation of curve $C$;  
$(II)$ If the parametric equation of line $l$ is $\begin{cases}x=1+ \frac{2}{\sqrt{5}}t \\ y=1+ \frac{1}{\sqrt{5}}t\end{cases}$ (where $t$ is the parameter), and point $P(1,1)$ is given, line $l$ intersects curve $C$ at points $A$ and $B$. Find the value of $|PA|+|PB|$.

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