
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 $\overrightarrow{a}=({\tan\theta,-1}),\overrightarrow{b}=({1,-2})$, where $\theta$ is an acute angle. If the angle between $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$ is $90^{\circ}$. Find the value of $\frac{{\sin({\pi-\theta})+\sin({\frac{{3\pi}}{2}+\theta})}}{{3\sin({\frac{\pi}{2}-\theta})-2\sin({\pi+\theta})}}$ and $\frac{{1-\cos2\theta}}{{\sin2\theta+\cos2\theta}}$.

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