
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
Let $f(x) = \sin(2x+\varphi)$ where $-\pi < \varphi < 0$. The graph of $y = f(x)$ has a symmetry axis at the line $x = -\frac{\pi}{12}$.  
$(1)$ Find the value of $\varphi$.  
$(2)$ Determine the interval where the function $y = f(x)$ is monotonically increasing.

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