
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
Find the minimum value for \(a, b > 0\) of the expression

\[
\frac{(2 a + 2 a b - b(b + 1))^{2} + \left(b - 4 a^{2} + 2 a (b + 1)\right)^{2}}{4 a^{2} + b^{2}}
\]

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