
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 $f(x) = \sin(\omega x + \phi)$ ($\omega > 0$, $0 \leq \phi \leq \pi$) is an even function, and the distance between a neighboring highest point and lowest point on its graph is $\sqrt{4+\pi^2}$.  
(1) Find the analytical expression of the function $f(x)$.  
(2) If $\sin\alpha + f(\alpha) = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan\alpha}$.

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