
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 $l$: 
$$
\begin{cases} 
x=1+ \frac {1}{2}t \\
y= \frac { \sqrt {3}}{2}t
\end{cases}
$$
($t$ is the parameter), and the curve $C_1$: $x^2+y^2=1$

(1) Suppose $l$ intersects $C_1$ at points $A$ and $B$, find $|AB|$.

(2) If the x-coordinates of the points on curve $C_1$ are compressed to half of their original values, and the y-coordinates are compressed to $\frac {\sqrt {3}}{2}$ of their original values, resulting in curve $C_2$, and let point $P$ be a moving point on curve $C_2$, find the minimum distance from $P$ to line $l$.

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