
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 the function $f(x) = ax^2 - 2ax + 1$ (where $a$ is a constant and $a > 0$). Consider the following statements:  
① The minimum value of the function $f(x)$ is $-1$;  
② The function $f(x)$ is monotonic on $\mathbb{R}$;  
③ If $f(x) > 0$ always holds, then the range of $a$ is $a > 1$;  
④ For any $x_1 < 0$, $x_2 < 0$ and $x_1 \neq x_2$, it always holds that $f(x_1) < f(x_2)$. The correct statement(s) is/are \_\_\_\_\_\_\_\_. (Write down all the correct statement numbers)

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