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

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