
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 $\sin 11^{\circ} \cos 19^{\circ} + \cos 11^{\circ} \sin 19^{\circ}$.
2. Find the smallest positive period of the function $f(x) = 3 \cos \left(\frac{\pi}{2} x - \frac{\pi}{8} \right)$.
3. If the terminal side of angle $\alpha$ passes through the point $P(-12, 5)$, find $\cos \alpha$.
4. If the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ is $\frac{2\pi}{3}$, and $|\mathbf{a}| = \sqrt{2}$, find the projection of $\mathbf{a}$ onto $\mathbf{b}$.
5. If $\tan \alpha$ and $\tan \beta$ are the two roots of the equation $x^2 - 3\sqrt{3}x + 4 = 0$, find $\tan (\alpha + \beta)$.

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