
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 that $\tan \alpha + \frac{1}{\tan \alpha} = \frac{10}{3}$, where $\alpha \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$, find the value of $\sin \left( 2\alpha + \frac{\pi}{4} \right) + 2 \cos \frac{\pi}{4} \cos^2 \alpha$.

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