
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
(1) Find the value of $\cos 89^{\circ}\cos 1^{\circ}+\sin 91^{\circ}\sin 181^{\circ}$.

(2) Find the value of $\tan 65^{\circ}+\tan 55^{\circ}-\sqrt{3}\tan 65^{\circ}\tan 55^{\circ}$.

(3) Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\dfrac{\pi}{3}$, and $|\overrightarrow{b}|=1$, $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{7}$, find $|\overrightarrow{a}|$.

(4) Given that $\tan\beta=\dfrac{4}{3}$ and $\sin(\alpha+\beta)=\dfrac{5}{13}$, where $\alpha,\beta\in(0,\pi)$, find the value of $\sin\alpha$.

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