
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 sine function $y = A \sin (\omega x + \varphi)$ with parameters $A > 0$, $\omega > 0$, and $0 < \varphi < 2 \pi$, where its graph intersects the y-axis at point $(0,1)$ and has a maximum point at $(2, \sqrt{2})$ within the same period, determine the expression of the function.

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