
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
$\triangle ABC$ and $\triangle ABD$ are inscribed in a circle of radius $r$ such that $AB$ is a chord within the circle, and $C$ and $D$ are on opposite arcs divided by $AB$. It's given that segment $CD$ is always perpendicular to $AB$. Points $C$ and $D$ do not coincide with $A$ or $B$. If $s = AC + AD$, then for all permissible positions of $C$ and $D$, find the maximum of $s^2$.

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