
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 that $A(x\_A, y\_A)$ is any point on the unit circle $(($center at the coordinate origin $O$, radius $= 1)$, rotate the ray $OA$ counterclockwise around point $O$ by $\frac{\pi}{6}$ to $OB$ intersecting the unit circle at point $B(x\_B, y\_B)$. If $m > 0$ and the maximum value of $m y\_A - 2 y\_B$ is $2$, find the value of the real number $m$.

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