
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 $\overrightarrow{a} = (2\cos x, \sqrt{2}\sin x - 1)$ and $\overrightarrow{b} = (\sin x, \sqrt{2}\sin x + 1)$, where $x \in \mathbb{R}$. Let the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$.

(Ⅰ) Find the maximum value of the function $f(x)$.

(Ⅱ) In $\triangle ABC$, angle $B$ is acute, and the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $f(B) = 1$, and the area of $\triangle ABC$ is $3$, with $a+c = 2 + 3\sqrt{2}$, find the value of $b$.

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