
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 \( a, b \in \mathbb{R} \) and \( ab = 1 \), find the minimum value of \( a^2 + b^2 \).
2. Given that \( a, b \in \mathbb{R}^+ \) and \( ab = 1 \), find the minimum value of \( a + b \).
3. Given that \( a, b \in \mathbb{R} \) and \( a^2 + b^2 = 1 \), find the maximum value of \( ab \).
4. Given that \( a, b \in \mathbb{R} \) and \( a + b = 1 \), find the maximum value of \( ab \).

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