
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 $f(x) = A\sin(\omega x + \varphi)$ ($A > 0$, $\omega > 0$, $0 < \varphi < \pi$) has a period of $\pi$, and it reaches its maximum value at $x = \frac{\pi}{12}$.  
(1) Find $\omega$ and $\varphi$;  
(2) If the equation $f(x) - 1 + A = 0$ has a real solution in the interval $\left[-\frac{\pi}{4}, 0\right]$, find the range of values for the real number $A$.

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