
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) The plane vectors $\vec{a}, \vec{b}, \vec{c}$ are not collinear, and the angles formed by any two of them are equal. If $|\vec{a}|=|\vec{b}|=2, |\vec{c}|=1$, then $|\vec{a}+\vec{b}+\vec{c}|=$ _______.

(2) In $\triangle ABC$, the angles $A, B, C$ are opposite to sides $a, b, c$ respectively, with $C=120^\circ$ and $a=2b$. Then, $\tan A=$ _______.

(3) Given the function $f(x)=x(m+e^{-x})$ (where $e$ is the base of the natural logarithm), there exist two distinct points on the curve $y=f(x)$ such that the tangents to the curve at these points are perpendicular to the $y$-axis. The range of the real number $m$ is _______.

(4) Given the function $f(x)=\cos x\sin 2x$, the correct conclusions among the following are _______  
① The graph of $y=f(x)$ is symmetric about the point $(\pi,0)$.  
② The graph of $y=f(x)$ is symmetric about the line $x=\frac{\pi}{2}$.  
③ The maximum value of $y=f(x)$ is $\frac{\sqrt{3}}{2}$.  
④ The function $y=f(x)$ is both an odd function and a periodic function.

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