
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
In the Cartesian coordinate system $xOy$, point $P\left( \frac{1}{2},\cos^2\theta\right)$ is on the terminal side of angle $\alpha$, and point $Q(\sin^2\theta,-1)$ is on the terminal side of angle $\beta$, and $\overrightarrow{OP}\cdot \overrightarrow{OQ}=-\frac{1}{2}$.  
$(1)$ Find $\cos 2\theta$;  
$(2)$ Find the value of $\sin(\alpha+\beta)$.

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