
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 $x_2$, identify the false statements among the following propositions:
(1) The equation may have two equal non-real roots.
(2) $ax^2+bx+c=(x-x_1)(x-x_2)$.
(3) $x_{1}^{2}x_{2}+x_{1}x_{2}^{2}=-\frac{bc}{a^2}$.
(4) If $b^2-4ac<0$, then $x_1-x_2$ is purely imaginary.

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