
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 \( a > 0 \) and \( b > 0 \), and \( a + b = 1 \), find the minimum value of \(\left(a + \frac{1}{a^{2}}\right)^{2} + \left(b + \frac{1}{b^{2}}\right)^{2}\). Also, find the maximum value of \(\left(a + \frac{1}{a}\right)\left(b + \frac{1}{b}\right)\).

(2) Given that \( a > 0 \), \( b > 0 \), and \( c > 0 \), and \( a + b + c = 1 \), find the minimum value of \( u = \left(a + \frac{1}{a}\right)^{3} + \left(b + \frac{1}{b}\right)^{3} + \left(c + \frac{1}{c}\right)^{3} \).

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