
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 vectors $\overrightarrow{a} = (\sqrt{2}\sin 2x, \sqrt{2}\cos 2x)$ and $\overrightarrow{b} = (\cos \theta, \sin \theta)$ ($|\theta| < \frac{\pi}{2}$), if $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$, and the graph of the function $f(x)$ is symmetric about the line $x = \frac{\pi}{6}$.
(Ⅰ) Find the analytic expression of function $f(x)$ and determine the interval where $f(x)$ is strictly decreasing;
(Ⅱ) In triangle $ABC$, the sides opposite to angle $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $f(A) = \sqrt{2}$, and $b = 5$, $c = 2\sqrt{3}$, find the area of the circumscribed circle of triangle $ABC$.

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