
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 that $α$ is an angle in the second quadrant, and $\tan α=\frac{1}{2}$, then $\cos α=$_______.

(2) Given $a=x^{α}$, $b=x^{ \frac{α}{2}}$, $c=x^{ \frac{1}{α}}$, where $x∈(0,1)$ and $α∈(0,1)$, the relationship among $a$, $b$, $c$ in terms of size is _______ (use “$<$” to connect).

(3) In $\triangle ABC$, $D$ is a point on side $BC$, $BC=3BD$, $AD=\sqrt{2}$, and $\angle ADB=135^{\circ}$. If $AC=\sqrt{2}AB$, then $BD=$_______.

(4) Definition: If function $f(x)$ is differentiable on $D$, meaning $f′(x)$ exists, and the derivative function $f′(x)$ is also differentiable on $D$, then $f(x)$ is said to have a second-order derivative on $D$, denoted as $f″(x)=(f′(x))′$. If $f″(x) < 0$ always holds on $D$, then $f(x)$ is called a convex function on $D$. Among the following four functions, which are convex functions on $\left(\begin{matrix}0, \frac{π}{2} \end{matrix}\right)$ are _______ (fill in all correct numbers).

①$f(x)=\sin x+\cos x$；②$f(x)=\ln x-2x$；③$f(x)=-x^{3}+2x-1$；④$f(x)=xe^{x}$．

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