
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 $f(x)=\cos ^{2}x-2\sin x\cos x-\sin ^{2}x$,
(1) Find the period and the interval(s) where $f(x)$ is increasing;
(2) If the graph of $f(x)$ is shifted to the left by $\frac{\pi}{8}$ to obtain the graph of function $g(x)$, find the range of $g(x)$ on $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$.

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