
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 $f(x)=\cos^2x+\cos^2\left(x-\frac{\pi}{3}\right)-1$, where $x\in \mathbb{R}$,

$(1)$ Find the smallest positive period and the intervals of monotonic decrease for $f(x)$;

$(2)$ The function $f(x)$ is translated to the right by $\frac{\pi}{3}$ units to obtain the function $g(x)$. Find the expression for $g(x)$;

$(3)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4},\frac{\pi}{3}\right]$;

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