
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) Given that $x > 0$, $y > 0$, $\frac{1}{x} + \frac{2}{y+1} = 2$, find the minimum value of $2x + y$.
(2) Given that $a > 0$, $b > 0$, $a + b = 1$, compare the magnitude of $8 - \frac{1}{a}$ and $\frac{1}{b} + \frac{1}{ab}$ and provide a reason.

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