
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
Let $f(x) = x - ax^2 + b$.
Let $f_0(x) = x^2 + b_0$, where $f(x) - f_0(\sin x)$ attains its maximum value $D$ on the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. 
When $a = 0$ and $b_0 = 0$, find the extremal value of $z = b - \frac{a^2}{4}$ under the condition $D \leq 1$.

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