
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 $f(x)=\cos(x+\varphi)$ $(-\pi < \varphi < 0)$, its derivative is $f'(x)$, and $g(x)=f(x)+f'(x)$ is an even function.

    (1) Find the value of $\varphi$;

    (2) Find the maximum value of the function $y=f(x) \cdot g(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$.

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