
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 a triangle $ABC$ where the sides opposite angles $A, B, C$ are $a, b, c$ respectively, and it's known that $$a\sin A + c\sin C - b\sin B = \sqrt{2}a\sin C.$$
$(1)$ Find the measure of angle $B$;
$(2)$ Suppose vector $\overrightarrow{m} = (\cos A, \cos 2A)$, vector $\overrightarrow{n} = (12, -5)$, and the side length $a = 4$. When the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ takes its maximum value, find the length of side $b$.

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