
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 the function $f(x) = 2\sin x \cdot \sin\left(x + \frac{\pi}{3}\right)$.  
(Ⅰ) Find the intervals of increase for the function $f(x)$.  
(Ⅱ) In an acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. The angle bisector of $A$ intersects $BC$ at $D$. The line $x = A$ is an axis of symmetry for the graph of the function $f(x)$, and $AD = \sqrt{2}BD = 2$. Find the length of side $a$.

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