
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
In the Cartesian coordinate system \( xOy \), the function \( f(x) = a \sin(ax) + \cos(ax) \) (where \( a > 0 \)) is defined. Find the area of the closed region formed by the graph of \( f(x) \) over its smallest positive period and the graph of \( g(x) = \sqrt{a^2 + 1} \).

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