
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
The following propositions are given: 
① The area of the closed region enclosed by the curve $y=x^2$ and the line $y=2x$ is $\frac{4}{3}$;
② Given point A is a fixed point on a given circle C, segment AB is a moving chord of the circle; if $\overrightarrow{OP} = \frac{1}{2}(\overrightarrow{OA}+\overrightarrow{OB})$, where O is the origin, then the locus of point P is a circle;
③ Distributing 5 distinct books to 4 people, with each person receiving at least 1 book, results in $A_5^4 \cdot A_4^1 = 480$ different ways;
④ If a line l is parallel to plane α, a line l is perpendicular to line m, and line l is contained in plane β, then β is perpendicular to α.
Among these, the correct propositions are __________. (Fill in all the correct proposition numbers on the line)

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