
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 space Cartesian coordinate system O-xyz, $\overrightarrow {OP} = x \overrightarrow {i} + y \overrightarrow {j} + z \overrightarrow {k}$ (where $\overrightarrow {i}, \overrightarrow {j}, \overrightarrow {k}$ are the unit vectors in the positive direction of the x-axis, y-axis, and z-axis, respectively). The following propositions are given:  
① If $\overrightarrow {OP} = x \overrightarrow {i} + y \overrightarrow {j} + 0 \overrightarrow {k} (x>0,y>0)$ and $|\overrightarrow {OP} - 4 \overrightarrow {j}| = |\overrightarrow {OP} + 2 \overrightarrow {i}|$, then the minimum value of $\frac {1}{x} + \frac {2}{y}$ is $2 \sqrt {2}$  
② If $\overrightarrow {OP} = 0 \overrightarrow {i} + y \overrightarrow {j} + z \overrightarrow {k}, \overrightarrow {OQ} = 0 \overrightarrow {i} + y_{1} \overrightarrow {j} + \overrightarrow {k}$, and vector $\overrightarrow {PQ}$ is collinear with $\overrightarrow {k}$ and $|\overrightarrow {PQ}| = |\overrightarrow {OP}|$, then the trajectory of the moving point P is a parabola;  
③ If $\overrightarrow {OM} = a \overrightarrow {i} + 0 \overrightarrow {j} + 0 \overrightarrow {k}, \overrightarrow {OQ} = 0 \overrightarrow {i} + b \overrightarrow {j} + 0 \overrightarrow {k}, \overrightarrow {OR} = 0 \overrightarrow {i} + 0 \overrightarrow {j} + c \overrightarrow {k} (abc \neq 0)$, then any point A(x, y, z) within plane MQR must satisfy the equation $\frac {x}{a} + \frac {y}{b} + \frac {z}{c} = 1$;  
④ Let $\overrightarrow {OP} = x \overrightarrow {i} + y \overrightarrow {j} + 0 \overrightarrow {k} (x \in [0,4],y \in [-4,4])$, $\overrightarrow {OM} = 0 \overrightarrow {i} + y_{1} \overrightarrow {j} + \overrightarrow {k} (y_{1} \in [-4,4])$, $\overrightarrow {ON} = x_{2} \overrightarrow {i} + 0 \overrightarrow {j} + 0 \overrightarrow {k} (x_{2} \in [0,4])$, if vector $\overrightarrow {PM} \perp \overrightarrow {j}$, $\overrightarrow {PN}$ is collinear with $\overrightarrow {j}$ and $|\overrightarrow {PM}| = |\overrightarrow {PN}|$, then the trajectory of the moving point P is part of a hyperbola.  
Among these, the correct propositions are \_\_\_\_\_\_\_\_\_\_\_\_.

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