
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
For the function $f(x)=\begin{cases}2x\cdot{e}^{x}, & x\leqslant 0 \\ x^2-2x+ \dfrac{1}{2}, & x > 0\end{cases}$, the following statements are made:

(1) At the point $(-2,f(-2))$ on the graph of the function, the slope of the tangent line is $- \dfrac{2}{{e}^{2}}$;

(2) The minimum value of the function $f(x)$ is $- \dfrac{2}{e}$;

(3) The graph of the function intersects the $x$-axis at $4$ points;

(4) The function $f(x)$ is decreasing on $(-\infty,1]$ and is also decreasing on $(0,1]$.

Among these statements, the correct ones are \_\_\_\_\_\_\_\_.

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