
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 the planar vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(3,1)$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.

(2) In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given $A=60^{\circ}$, $a=\sqrt{31}$, and $b=6$, find the value of $c$.

(3) Given $f(x)=\begin{cases}
\sin \frac{\pi }{2}x, & x\leqslant 0 \\
f(x-2)+\frac{3}{2}, & x > 0
\end{cases}$, find the value of $f(\frac{5}{3})$.

(4) The function $f(x)$ ($x\in R$) satisfies $f(-x)=4-f(x)$. The function $g(x)=\frac{x-2}{x-1}+\frac{x}{x+1}$. If the intersection points of the curve $y=f(x)$ and $y=g(x)$ are $(x\_1, y\_1)$, $(x\_2, y\_2)$, $(x\_3, y\_3)$, ..., $(x\_m, y\_m)$, express the sum $\sum\_{i=1}^{m}{(x\_i+y\_i)}$ using a formula containing $m$.

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