
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 vectors $\overrightarrow{a}=(x,x+1)$ and $\overrightarrow{b}=(1,2)$, and $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $x$.

(2) Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are two non-collinear vectors, and $\overrightarrow{{AB}}=2\overrightarrow{e_{1}}+k\overrightarrow{e_{2}}$, $\overrightarrow{{CB}}=\overrightarrow{e_{1}}+3\overrightarrow{e_{2}}$, $\overrightarrow{{CD}}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the value of $k$ if points $A$, $B$, and $D$ are collinear.

(3) In $\triangle ABC$, angles $A,B,C$ have corresponding sides $a,b,c$. If $(\sqrt{3}b-c)\cos A=a\cos C$, find the value of $\cos A$.

(4) In $\triangle ABC$, angles $A,B,C$ have corresponding sides $a,b,c$. If $a^{2}-b^{2}=\sqrt{3}bc$ and $\sin C=2\sqrt{3}\sin B$, find the value of angle $A$.

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