
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 $f(x) = \sin(2\omega x + \varphi) - 1$ ($\omega > 0$, $|\varphi| < \frac{\pi}{2}$), its smallest positive period is $\frac{\pi}{2}$, and the graph passes through the point $(0, -\frac{1}{2})$.

$(1)$ Find the values of $\omega$ and $\varphi$, and the intervals of monotonic increase for $f(x)$;

$(2)$ If the graph of function $f(x)$ is translated to the right by $\frac{\pi}{8}$ units, and then the x-coordinates of all points on the graph are stretched to twice their original values (the y-coordinates remain unchanged), resulting in the graph of function $y = g(x)$, and if the function $F(x) = g(x) + k$ has exactly two distinct zeros in the interval $\left[0, \frac{\pi}{2}\right]$, find the range of values for the real number $k$.

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