
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 $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $0 < \varphi < \frac{\pi}{2}$), the graph intersects with the x-axis, and the distance between two adjacent intersection points is $\frac{\pi}{4}$. Also, the graph passes through point M ($\frac{\pi}{3}, -1$).  
(1) Find the expression of $f(x)$.  
(2) Find the intervals of monotonic increase for $f(x)$.  
(3) Move the graph of $f(x)$ to the right by $\frac{\pi}{8}$ units, and then stretch the x-coordinates of all points on the graph by a factor of 2 (keeping the y-coordinates unchanged) to obtain the graph of the function $y=g(x)$. If the equation $g(x) + k = 0$ has exactly one real solution in the interval $[0, \frac{\pi}{2}]$, find the range of values for the real number $k$.

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