
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 $f\left(\theta \right)=\frac{sin(\theta+\frac{5}{2}\pi)cos(\frac{3}{2}\pi-\theta)cos(\theta+3\pi)}{cos(-\frac{\pi}{2}-\theta)sin(-\frac{3}{2}\pi-\theta)}$.  
$(1)$ Simplify $f\left(\theta \right)$;  
$(2)$ If $\sin (\theta -\frac{\pi}{6})=\frac{3}{5}$, find the value of $f(\theta +\frac{\pi}{3})$.

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