
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
Consider the domain  $D$  expressed by the following system inequality:
\[x^2+(y-1)^2\leq 1,\ x\geq \frac{\sqrt{2}}{3}.\]

Suppose that the line  $l$  passes through the origin and the common part of  $l$  and  $D$  is line segment.
Find the maximum length of the line segment  $L$ , then when  $L$  is maximized, find the value of  $\cos \theta$  for the angle  $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$  between the  $x$ -axis and  $l$ .

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