
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 a quadratic equation with real coefficients $ax^{2}+bx+c=0$ whose roots are $x_{1}$ and $x_{2}$, among the following statements, the false statement(s) is/are __________  
$(1)$ The equation may have two equal imaginary roots  
$(2)$ $ax^{2}+bx+c=(x-x_{1})(x-x_{2})$  
$(3)$ $x_{1}^{2}x_{2}+x_{1}x_{2}^{2}=-\dfrac{bc}{a^{2}}$  
$(4)$ If $b^{2}-4ac < 0$, then $x_{1}-x_{2}$ is definitely a pure imaginary number.

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