
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
If the functions $f(x)$ and $g(x)$ satisfy $\int_{-1}^{1}f(x)g(x)dx=0$, determine the number of sets of the given functions that are orthogonal on the interval $[-1,1]$: $(1) f(x)=\sin \frac{1}{2}x$, $g(x)=\cos \frac{1}{2}x$; $(2) f(x)=x+1$, $g(x)=x-1$; $(3) f(x)=x$, $g(x)=x^{2}$.

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