
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
$D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\angle ABC=\alpha$ and $\angle CAB=\beta$. <br/>$(1)$ When $CD\perp AB$ and $\beta =2\alpha$, find the value of $CD$; <br/>$(2)$ If $α+\beta=\frac{π}{4}$, find the maximum area of triangle $ACD$.

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