
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
For a plane vector $\overrightarrow {a}$ with coordinates (x, y), we define its "new magnitude" as $|x+y|+|x-y|$, which is still denoted by $|\overrightarrow {a}|$, that is, $|\overrightarrow {a}|=|x+y|+|x-y|$. Under this definition of "new magnitude," consider the following propositions:  
(1) For any two vectors $\overrightarrow {a}, \overrightarrow {b}$ in the plane, it always holds that $|\overrightarrow {a}- \overrightarrow {b}|\leq|\overrightarrow {a}|+|\overrightarrow {b}|$;  
(2) Let O be the coordinate origin and point P move on the line $y=x-1$, then the minimum value of $|\overrightarrow {OP}|$ is 1;  
(3) Let O be the coordinate origin and point P move on the circle $O: x^2+y^2=1$, then the maximum value of $|\overrightarrow {OP}|$ is 2;  
(4) Let O be the coordinate origin and point P move on the ellipse $\frac {x^{2}}{4}+ \frac {y^{2}}{1}=1$, then the minimum value of $|\overrightarrow {OP}|$ is 2;  
Write down the number(s) of all the correct propositions ______ .

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