
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
Among the following statements:
① If $f(x) = ax^2 + (2a+b)x + 2$ (where $x \in [-1, a]$) is an even function, then the real number $b = -2$;
② $f(x) = \sqrt{2016 - x^2} + \sqrt{x^2 - 2016}$ is both an odd and an even function;
③ If $f(x+2) = \frac{1}{f(x)}$ when $x \in (0, 2)$, and $f(x) = 2^x$, then $f(2015) = 2$;
④ Given $f(x)$ is defined on $\mathbb{R}$ and is not identically zero, and for any $x, y \in \mathbb{R}$, it satisfies $f(xy) = xf(y) + yf(x)$, then $f(x)$ is an odd function. The correct statement numbers are __________.

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