
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 the function $f(x) = \sin wx + \partial (w > 0, \partial < \frac{\pi}{2})$, find the monotonically decreasing interval in one of its periods.
(1) Find the analytical expression of $f(x)$;
(2) First shift the graph of $y=f(x)$ to the right by $\frac{\pi}{6}$ units, then change the horizontal coordinates of all points on the graph to half of their original values (the vertical coordinates remain unchanged). The function corresponding to the resulting graph is denoted as $g(x)$. If for any $x \in [\frac{\pi}{8}, \frac{3\pi}{8}]$, the inequality $|y(x) - m| < 1$ always holds, find the range of values for the real number $m$.

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