
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 $f(x) = \sin x \cos x \sin \varphi + \cos^2 x \cos \varphi + \frac{1}{2} \cos (\pi + \varphi)$ $(0 < \varphi < \pi)$, its graph passes through the point $\left( \frac{\pi}{3}, \frac{1}{4} \right)$.  
(1) Find the value of $\varphi$;  
(2) If the graph of the function $y=f(x)$ is translated left by $\frac{\pi}{6}$ units to obtain the graph of the function $y=g(x)$, find the intervals of increase for the function $g(x)$ on $\left[ -\frac{\pi}{4}, \frac{2\pi}{3} \right]$.

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