
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)
i) Evaluate the expression $\frac{2\sin {{46}^{\circ }}-\sqrt{3}\cos {{74}^{\circ }}}{\cos {{16}^{\circ }}}=$ \_\_\_\_\_\_\_\_\_

ii) Evaluate the expression $\sin 42{}^\circ \cos 18{}^\circ -\cos 138{}^\circ \cos 72{}^\circ =$\_\_\_\_\_\_\_\_

(2)
i) Let the function $f(x)=\begin{cases} & x,x < 1 \\ & {{x}^{3}}-\frac{1}{x}+1,x\geqslant 1 \\ \end{cases}$, find the solution set of the inequality $f(6-{{x}^{2}}) > f\left( x \right)$.

ii) Let the function $f(x)=\begin{cases} & x,x < 1 \\ & {{x}^{3}}-\frac{1}{x}+1,x\geqslant 1 \\ \end{cases}$, evaluate $f(\dfrac{1}{f(2)}) =$\_\_\_\_\_\_\_\_\_\_

(3)
i) Let the function $f(x)=\sin (3x+ \frac{π}{4})$, find the minimum value of $m(m > 0)$ such that the function $f(x)$ translated $m$ units to the left is an even function.

ii) Find the minimum positive period of the function $f(x)=\sin (3x+ \frac{π}{4})$.

(4)
i) In isosceles triangle $\Delta ABC$, the vertex angle $A=\frac{2π }{3}$, and $\left| BC \right|=2\sqrt{3}$. Taking $A$ as the center and $1$ as the radius, draw a circle with $PQ$ as the diameter, find the maximum value of $\overrightarrow{BP}\cdot \overrightarrow{CQ}$.

ii) In isosceles triangle $\Delta ABC$, the vertex angle $A=\frac{2π }{3}$, and $\left| BC \right|=2\sqrt{3}$, find the value of $\overrightarrow{BA}\bullet \overrightarrow{AC}$.

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