
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
If the function $f(x) = 2\sin(x+\varphi)$ where $(0 < \varphi < \pi)$, and $f(x) + f'(x)$ is an even function.  
$(1)$ Find the value of $\varphi$;  
$(2)$ Let $g(x) = f(x) + x$, $x \in (0, 2\pi)$. Find the monotonic intervals of $g(x)$.

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