
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 triangle $ABC$ with side lengths of $3$, $5$, and $7$, the radius of the circumcircle of this triangle is _______.

(2) In $\triangle ABC$, if $\angle A= \frac{2\pi}{3}$ and $a= \sqrt{3}c$, then $\frac{b}{c} =$ _______.

(3) Observing the sequence $\frac{3}{2}, -\frac{8}{3}, \frac{15}{4}, -\frac{24}{5}, \frac{35}{6}, -\frac{48}{7}, \ldots$, a general formula for its terms is _______.

(4) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $b^{2}+c^{2}=a^{2}-bc$ and $\vec{AC} \cdot \vec{AB}=-4$, then the area of $\triangle ABC$ is _______.

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