
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: $\dfrac{\sin \theta}{a}x + \dfrac{\cos \theta}{b}y = 1$ ($a,b$ are given positive constants, $\theta$ is a parameter, $\theta \in [0,2\pi)$) forms a set $S$. Consider the following propositions:

$(1)$ When $\theta = \dfrac{\pi}{4}$, the slope of the lines in $S$ is $\dfrac{b}{a}$;

$(2)$ All lines in $S$ can cover the entire coordinate plane;

$(3)$ When $a=b$, there exists a fixed point, the distance from this fixed point to all lines in $S$ is equal;

$(4)$ When $a > b$, the minimum distance between two parallel lines in $S$ is $2b$;

Among these, the correct propositions are __________ (Write down the numbers of all correct propositions).

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