
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 $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\cos x, \frac{3}{2})$, and the function $f(x)=( \overrightarrow{m}+ \overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing.
(2) The graph of $f(x)$ is translated to the left by $\frac{\pi}{8}$ units to obtain the graph of $g(x)$. In triangle $ABC$, if the lengths of the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $a=3$, $g\left(\frac{A}{2}\right)=\frac{\sqrt{6}}{6}$, and $\sin B=\cos A$, find the value of $b$.

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