
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 zero of the function $f(x)=- \frac {1}{x}+\lg x$ is in the interval $(2,3)$;
2. The equation of the tangent line to the curve $y=4x-x^3$ at the point $(-1,-3)$ is $y=x-2$;
3. Translating the graph of the function $y=2^x+1$ by the vector $\vec{a}=(1,-1)$ results in the graph of the function $y=2^{x+1}$;
4. The domain of the function $y= \sqrt{\log_{\frac{1}{2}}(x^2-1)}$ is $(-\sqrt{2},-1)\cup(1,\sqrt{2})$;
5. $\vec{a}\cdot\vec{b}>0$ is a necessary and sufficient condition for the angle between $\vec{a}$ and $\vec{b}$ to be acute;

The correct statements are __________.

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