
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 $f(x) = A\sin(\omega x + \varphi) + 1$ ($A > 0$, $\omega > 0$, $0 < \varphi < \frac{\pi}{2}$), the maximum value of the function $f(x)$ is 3, the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$, and $f(0) = 2$.

(1) Find the expression for the function $f(x)$;  
(2) The graph of function $f(x)$ is translated to the left by $\frac{\pi}{6}$ units and then translated downward by 1 unit to obtain the graph of function $g(x)$. Determine the parity of $g(x)$ and find the intervals of monotonic increase for $g(x)$ on $\mathbb{R}$.

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