
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
Let $a > 0$, $b > 0$, and $a+b=2$.

$(1)$ Find the maximum value of $a \cdot b$;  
$(2)$ Find the minimum value of $\dfrac{2}{a} + \dfrac{8}{b}$.

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