
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
For the quadratic function $y=x^2-4x+a$, consider the following statements:  
(1) When $x<1$, $y$ decreases as $x$ increases;  
(2) If the graph intersects the x-axis, then $a\leq 4$;  
(3) When $a=3$, the range of $x$ for which $y=x^2-4x+a$ is greater than $0$ is $1<x<3$;  
(4) If $x=2013$ results in $y=b$, then $x=-2009$ also results in $y=b$.  
Among these statements, which ones do you think are correct? Please fill in the numbers of the correct statements.

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