
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
Given the real numbers \(a\) and \(b\) satisfying \(\left(a - \frac{b}{2}\right)^2 = 1 - \frac{7}{4} b^2\), let \(t_{\max}\) and \(t_{\min}\) denote the maximum and minimum values of \(t = a^2 + 2b^2\), respectively. Find the value of \(t_{\text{max}} + t_{\text{min}}\).

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