
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
Determine the true statement(s) from the following four propositions:
① If $x_1$ and $x_2$ are the two real roots of the quadratic equation $ax^2+bx+c=0$, with $x_1<x_2$, then the solution set for the inequality $ax^2+bx+c<0$ is $\{x|x_1<x<x_2\}$;
② If $\frac{x-1}{x-2}\le0$, then $(x-1)(x-2)\le0$;
③ The contrapositive statement of "If $m>2$, then the solution set for $x^2-2x+m>0$ is the set of all real numbers $\mathbb{R}$";
④ For a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(-x)=-f(x)$ for all $x\in \mathbb{R}$ and $f(1+x)=f(1-x)$, then $4$ is a period for $y=f(x)$.
Fill in the blank with the correct proposition number(s) you believe to be true.

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