
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 $f(x) = \sqrt{2}\sin(2\omega x + \varphi)$ ($x \in \mathbb{R}$), where $\omega > 0$, $|\varphi| < \frac{\pi}{2}$, and $f(x)$ satisfies the following two conditions: ① The distance between two adjacent axes of symmetry is $\pi$; ② $f(0) = 1$.  
(Ⅰ) Find the expression of the function $f(x)$;  
(Ⅱ) Find the interval of monotonic increase of $f(x)$ in $[0, \pi]$;  
(Ⅲ) If the equation $f(x) + a = 0$ has 2 distinct real roots in $[0, \frac{5\pi}{8}]$, find the range of the real number $a$.

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