
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 a geometric sequence with positive terms $\{a_n\}$, where $a_1=1$, and the sum of the first $n$ terms is $S_n (n \in N^*)$, and $\frac{1}{a_1} - \frac{1}{a_2} = \frac{2}{a_3}$, then $S_4=$ _______.

(2) Let $\omega > 0$, the graph of the function $y=\sin (\omega x+\frac{\pi }{3})+2$ coincides with its original image after being shifted to the right by $\frac{4\pi }{3}$ units, then the minimum value of $\omega$ is _______.

(3) For a school art festival, among four types of entries A, B, C, and D, only one first prize is awarded. Before the announcement of the awards, four students made predictions about which entry would win the first prize as follows:

- A said: "Entry C or D will win the first prize."
- B said: "Entry B will win the first prize."
- C said: "Entries A and D will not win the first prize."
- D said: "Entry C will win the first prize."

If only two of these students are correct, then the entry that won the first prize is _______.

(4) In $\Delta ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $a > b, a > c$. The radius of the circumcircle of $\Delta ABC$ is $1$, and $a=\sqrt{3}$. If a point $D$ on side $BC$ satisfies $\overrightarrow{BD}=3\overrightarrow{DC}$, and $\angle BAD=90^{\circ}$, then the area of $\Delta ABC$ is _______.

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