
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
(1) A ray of light is emitted from point $A(1,0)$ and reflects off a point $B(0,2)$ on the $y$-axis. What is the equation of the line that the reflected ray lies on?

(2) What is the interval of increase for the function $y=2\sin \left(2x- \frac{\pi}{3}\right)$?

(3) Given a function $f(x)$ defined on $\mathbb{R}$, whose graph is symmetric about the point $(-\frac{3}{4},0)$ and satisfies $f(x)=-f(x+\frac{3}{2}), f(-1)=1$, and $f(0)=-2$, find the sum $f(1)+f(2)+\ldots+f(2010)$.

(4) Let the function $f(x)=|2^{x}-1|$ have both its domain and range as $[a,b]$. Find $a+b$.

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