
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: $2^{-1}+|\sqrt{6}-3|+2\sqrt{3}\sin 45^{\circ}-\left(-2\right)^{2023}\cdot (\frac{1}{2})^{2023}$.  
$(2)$ Simplify and then evaluate: $\left(\frac{3}{a+1}-a+1\right) \div \frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-1$, $2$, $3$ for evaluation.

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