
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 the function $f(x)=\sin(x+\varphi)$ where $(0 < \varphi < \frac{\pi}{2})$, and the graph of $y=f(x)$ has a line of symmetry which is the line $x= \frac{\pi}{4}$.  
$(1)$ Find $\varphi$;  
$(2)$ Find the intervals where the function $y=f(x)$ is increasing.

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