
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 following propositions, the true ones are.  
(1) In the complex plane, the trajectory of the complex number $z$ satisfying $|z-2|-|z+2|=1$ is a hyperbola;  
(2) When $a$ varies in the set of real numbers $\mathbb{R}$, the trajectory of the complex number $z=a^2+ai$ in the complex plane is a parabola;  
(3) Given the function $y=f(x)$, $x\in\mathbb{R}^+$ and the sequence $a_n=f(n)$, $n\in\mathbb{N}$, then "the sequence $a_n=f(n)$, $n\in\mathbb{N}$ is increasing" is a necessary but not sufficient condition for "the function $y=f(x)$, $x\in\mathbb{R}^+$ is increasing";  
(4) In the Cartesian coordinate system $xoy$, translating the curve corresponding to the equation $g(x,y)=0$ by the vector $(1,2)$ results in the new curve's equation being $g(x-1,y-2)=0$;  
(5) Suppose in the Cartesian coordinate system $xoy$, the equation $F(x,y)=0$ represents an ellipse, then there always exist real constants $p$, $q$ such that the equation $F(px,qy)=0$ represents a circle.

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