
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
$(1)$ Calculate: $|3-\sqrt{12}|+(\frac{1}{3})^{-1}-4\sin 60^{\circ}+(\sqrt{2})^{2}$.  
$(2)$ Find the solution set of the inequality system $\left\{\begin{array}{l}{5x-1＜3(x+1)①}\\{\frac{x+1}{2}≥\frac{2x+1}{5}②}\end{array}\right.$.  
$(3)$ Simplify first, then find the value of $\left(\frac{1}{x+1}+x-1\right)\div \frac{{x}^{2}}{{x}^{2}+2x+1}$, where $x$ is the root of the equation $x^{2}-2x-3=0$.

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