
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 $x \in (0, \frac{\pi}{2})$, and the minimum value of the function $f(x) = \frac{1 + 2\sin^2 x}{\sin 2x}$ is $b$. If the function $g(x) = \begin{cases} -1 & (\frac{\pi}{4} < x < \frac{\pi}{2}) \\ 8x^2 - 6bx + 4 & (0 < x \leqslant \frac{\pi}{4}) \end{cases}$, determine the solution set of the inequality $g(x) \leqslant 1$.

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