
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 $\overrightarrow {a}=(\sin(2x+ \frac {\pi}{6}), 1)$, $\overrightarrow {b}=( \sqrt {3}, \cos(2x+ \frac {\pi}{6}))$, and the function $f(x)= \overrightarrow {a} \cdot \overrightarrow {b}$.

(Ⅰ) Find the interval where the function $f(x)$ is monotonically decreasing;  
(Ⅱ) In $\triangle ABC$, where $A$, $B$, and $C$ are the opposite sides of $a$, $b$, and $c$ respectively, if $f(A)= \sqrt {3}$, $\sin C= \frac {1}{3}$, and $a=3$, find the value of $b$.

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