
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
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\rho\cos \theta+\rho\sin \theta=1$, and the polar equation of curve $C$ is $\rho\sin^2\theta=8\cos \theta$.

$(1)$ Find the Cartesian equations of line $l$ and curve $C$;  
$(2)$ Let point $M(0,1)$, line $l$ intersects curve $C$ at two different points $P$, $Q$, find the value of $|MP|+|MQ|$.

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