
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 $f(x)= \sqrt{3}\sin x\cos x+\sin^2x$.

(Ⅰ) Find the increasing interval of the function $f(x)$.

(Ⅱ) In 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 $D$, $f(A)= \frac{3}{2}$, $AD= \sqrt{2}BD=2$, find $\cos c$.

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