
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 $f(x) = \sin(\omega x + \phi) - \cos(\omega x + \phi)$, where $(0 < \phi < \pi, \omega > 0)$, is an even function, and the distance between two adjacent axes of symmetry of the graph of $y = f(x)$ is $\frac{\pi}{\omega}$.  
(1) Find the value of $f$;  
(2) After translating the graph of $y = f(x)$ to the right by $\frac{\pi}{4\omega}$ units, and then stretching the x-coordinates of the points on the obtained graph by 4 times while keeping the y-coordinates unchanged, we get the graph of the function $y = g(x)$. Find the expression for $g(x)$ and its intervals of monotonic decrease.

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