
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 curve $C$: $\frac {x^{2}}{4-k}+ \frac {y^{2}}{k-1}=1$, provide the correct propositions among the following:
$①$ The curve $C$ cannot represent an ellipse;
$②$ If the curve $C$ represents a hyperbola, then $k < 1$ or $k > 4$;
$③$ When $1 < k < 4$, the curve $C$ represents an ellipse;
$④$ If the curve $C$ represents an ellipse with foci on the $x$-axis, then $1 < k < \frac {5}{2}$.
The correct proposition(s) is/are \_\_\_\_\_\_.

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