
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. If the acute angles $α$ and $β$ satisfy $(1+ \sqrt{3} \tan α)(1+ \sqrt{3} \tan β)=4$, find the value of $α+β$.

2. Given that $2 \overset{→}{a} - \overset{→}{b} =(-1, \sqrt{3} )$, $\overset{→}{c} =(1, \sqrt{3} )$, $\overset{→}{a} · \overset{→}{c} =3$, and $|\overset{→}{b} |=4$, find the angle between $\overset{→}{b}$ and $\overset{→}{c}$.

3. Given that $x ∈R$, $\overset{→}{AB} =(-1,x+2)$, and $\overset{→}{CD} =(x,1)$, find the maximum value of the projection of $\overset{→}{CD}$ on the direction of $\overset{→}{AB}$.

4. Given the function $f(x)=\sin^2 x+\sin x\cos x-\frac{1}{2}$, among the following conclusions:
   (1) The function $f(x)$ is symmetric about $x=\frac{π}{8}$;
   (2) The function $f(x)$ is symmetric about $(-\frac{π}{8},0)$;
   (3) The function $f(x)$ is increasing in $(0, \frac{π}{8})$;
   (4) Translating the graph of $y=\frac{\sqrt{2}}{2} \cos 2x$ to the right by $\frac{3π}{8}$ gives the graph of $f(x)$.
   Determine the correct conclusion number(s).

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