
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
If the curve represented by the equation $\dfrac{x^2}{4-t} + \dfrac{y^2}{t-1} = 1$ is denoted by $C$, consider the following four propositions:  
$(1)$ If $C$ is an ellipse, then $1 < t < 4$;  
$(2)$ If $C$ is a hyperbola, then $t > 4$ or $t < 1$;  
$(3)$ The curve $C$ cannot be a circle;  
$(4)$ If $C$ represents an ellipse, and its major axis is on the $x$-axis, then $1 < t < \dfrac{3}{2}$.  
Among these propositions, the correct ones are __________ (Fill in all correct proposition numbers).

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