
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 $f(x)=\sin x + \sin\left(x+ \frac{\pi}{2}\right)$, with $x\in\mathbb{R}$,
1. Find the smallest positive period of $f(x)$.
2. Find the maximum value of $f(x)$ and the corresponding set of values for $x$.
3. If $f(\alpha)= \frac{3}{4}$, find the value of $\sin 2\alpha$.

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