
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, point $A(\cos \theta, \sqrt{2}\sin \theta), B(\sin \theta, 0)$ where $\theta \in \mathbb{R}$.
(1) When $\theta \in [0, \frac{\pi}{2}]$, find the maximum value of $|\overrightarrow{AB}|$.
(2) When $\theta \in [0, \frac{\pi}{2}]$ and $|\overrightarrow{AB}|= \sqrt{\frac{5}{2}}$, find the value of $\sin(2\theta + \frac{5\pi}{12})$.

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