
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
Among the four propositions given below:

1. For the parabola $y^{2}=4x$ with the focus as the center and passing through the coordinate origin, the equation of the circle is $(x-1)^{2}+y^{2}=1$;
2. If $m=-2$, then the line $(m+2)x+my+1=0$ is perpendicular to the line $(m-2)x+(m+2)y-3=0$;
3. The negation of the proposition "There exists an $x \in \mathbb{R}$ such that $x^{2}+3x+4=0$" is "For all $x \in \mathbb{R}$, $x^{2}+3x+4 \neq 0$";
4. Shifting the graph of the function $y=\sin{2x}$ to the right by $\frac{\pi}{3}$ units results in the graph of the function $y=\sin{(2x-\frac{\pi}{6})}$.

Among these, the true propositions are ______ (fill in the correct serial numbers).

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