
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 that $a$ is any real number in the interval $[0, 1]$, and the functions are $f_1(x) = x-a$, $f_2(x) = -x^2 + 1$, and $f_3(x) = -x^3 + x^2$, then evaluate the following statements:
① For any $x_0 \in \mathbb{R}$, there exist $f_i(x)$ and $f_j(x)$ (where $\{i,j\} \subset \{1,2,3\}$) such that $f_i(x)f_j(x) \geq 0$;
② For any $x_0 \in \mathbb{R}$, there exist $f_i(x)$ and $f_j(x)$ (where $\{i,j\} \subset \{1,2,3\}$) such that $f_i(x)f_j(x) \leq 0$;
③ For any functions $f_i(x)$ and $f_j(x)$ (where $\{i,j\} \subset \{1,2,3\}$), there exists $x_0 \in \mathbb{R}$ such that $f_i(x)f_j(x) > 0$;
④ For any functions $f_i(x)$ and $f_j(x)$ (where $\{i,j\} \subset \{1,2,3\}$), there exists $x_0 \in \mathbb{R}$ such that $f_i(x)f_j(x) < 0$.

The correct statements are __________. (Fill in all the correct statement numbers)

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