
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 \( a > 0 \)) has a graph over an interval of its smallest positive period. Determine the area of the closed region enclosed by the graph of the function \( f(x) \) and the graph of the function \( g(x) = \sqrt{a^2 + 1} \).

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