
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 $f(x)=\sin x(\cos x- \sqrt {3}\sin x)$.
(1) Find the intervals where the function $f(x)$ is strictly increasing on $[0,π]$.
(2) In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively. If $f(B)=0$ and $a$, $b$, and $ \sqrt {3}c$ form an arithmetic sequence with a positive common difference, find the value of $\frac{\sin A}{\sin C}$.

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