
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) Given $\cos \left(\frac{\pi}{3} + \alpha\right) = \frac{1}{3}$ $(0 < \alpha < \frac{\pi}{2})$, find $\sin (\pi + \alpha) = $ ______.

(2) Let $x$ and $y$ satisfy the constraints $\begin{cases} x+3y \leqslant 3, \\ x-y \geqslant 1, \\ y \geqslant 0, \end{cases}$, the maximum value of $Z=|x|+y$ is: ______.

(3) If the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}+2ay-6=0$ $(a > 0)$ have a common chord of length $2\sqrt{3}$, then $a=$ ______.

(4) Five students stand in a circle and count in sequence, with the rules:

① The first student starts counting from $1$. The second student also starts from $1$, and thereafter, each student announces a number that is the sum of the numbers announced by the previous two students;

② If the number announced is a multiple of $3$, the student must clap once. When the $30^{th}$ number is announced, the total number of claps by the five students is ______.

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