
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 Example 1 (2), we construct a trigonometric problem $1 < \sin \alpha + \cos \alpha < \frac{\pi}{2}, \alpha \in \left(0, \frac{\pi}{2}\right)$. However, through algebraic operations, we can derive a stronger conclusion $\sin \alpha + \cos \alpha = \sqrt{2} \sin \left( \alpha + \frac{\pi}{4} \right) \leqslant \sqrt{2}$ (with equality when $\alpha = \frac{\pi}{4}$). Construct this problem using the unit circle.

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