
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 $f(x)=4\sin x\sin \left(x+ \frac {\pi}{3}\right)$, in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively.  
(Ⅰ) Find the monotonic increasing interval of the function $f(x)$.  
(Ⅱ) If $f(A)=3$ and $b+c=6$, find the minimum value of $a$.

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