
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
The graph of the function $f(x) = \sin 2x$ is translated to the right by $\varphi \left( 0 < \varphi < \frac{\pi }{2} \right)$ units to obtain the graph of the function $g(x)$. If the function $g(x)$ is monotonically increasing in the interval $\left[ 0,\frac{\pi }{3} \right]$ and the largest negative zero of the function $g(x)$ is in the interval $\left( -\frac{\pi }{3},-\frac{\pi }{12} \right)$, determine the range of $\varphi$.

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