
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: $-1^{2}+\sqrt[3]{-8}-|1-\sqrt{2}|+\sqrt{(-2)^{2}}$.  
$(2)$ Solve the system of inequalities: $\left\{\begin{array}{l}{x-3(x-2) \geq 4①}\\{\frac{2x-1}{5} < \frac{x+1}{2}②}\end{array}\right.$, and write down its non-negative integer solutions.

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