
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 line $l$: $\begin{cases} & x=\sqrt{3}t \\ & y=2-t \end{cases}$, ($t$ as the parameter) and the parabola $C$: $y^{2}=2x$, line $l$ intersects parabola $C$ at points $P_{1}$ and $P_{2}$. The sum of the distances from point $A(0,2)$ to points $P_{1}$ and $P_{2}$ is ______.

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