
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 vector $\mathbf{a}=(2,-1)$ and vector $\mathbf{b}=(m,3)$. If $\mathbf{a} \parallel \mathbf{b}$, then the value of $m$ is _______.

(2) Given an angle $\alpha$ with a point $P(-4,3)$ on its terminal side, then $\sin 2\alpha =$ _______.

(3) Regarding the following statements:

① The tangent function $y=\tan x$ is an increasing function within its domain;

② The function $f(x)=\cos \left( \frac{2}{3}x+ \frac{\pi}{2}\right)$ is an odd function;

③ $x= \frac{\pi}{8}$ is an equation of a line of symmetry for the function $f(x)=\sin \left(2x+ \frac{5\pi}{4}\right)$;

④ If the perimeter of a sector is $8\,cm$ and its area is $4\,cm^2$, then the central angle of the sector is $2\,rad$;

Among these, the correct ones are _______. (Write down all correct answer numbers)

(4) Given the function $f(x)=x-\frac{1}{2}ax^2-\ln (1+x)$ has a maximum value of $0$ on the interval $[0,+\infty)$, then the range of the real number $a$ is _______.

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