
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 directed distance from a point $P(x_{0},y_{0})$ to a line $l$: $Ax+By+C=0(A^{2}+B^{2}	eq 0)$ is defined as $d= 	rac {Ax_{0}+By_{0}+C}{ 	rac {A^{2}+B^{2}}}.$ Given points $P_{1}$, $P_{2}$ with directed distances $d_{1}$, $d_{2}$ to line $l$, evaluate the following propositions:

$	o{1}$ If $d_{1}=d_{2}$, then line $P_{1}P_{2}$ is parallel to line $l$;

$	o{2}$ If $d_{1}=-d_{2}$, then line $P_{1}P_{2}$ is perpendicular to line $l$;

$	o{3}$ If $d_{1}cdot d_{2} > 0$, then line $P_{1}P_{2}$ intersects or is parallel to line $l$;

$	o{4}$ If $d_{1}cdot d_{2} < 0$, then line $P_{1}P_{2}$ intersects line $l$.

Provide the sequence numbers of all correct propositions.

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