
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 a Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the initial line, a polar coordinate system is established. The polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$.  
$(1)$ Find the Cartesian equation of curve $C$;  
$(2)$ The parametric equation of line $l$ is $\begin{cases} x=1+ \frac{2\sqrt{5}}{5}t \\ y=1+ \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is the parameter), let point $P(1,1)$, and line $l$ intersects with curve $C$ at points $A$, $B$. Calculate the value of $|PA|+|PB|$.

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