
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 triangle $\triangle ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. It satisfies the equation $\sqrt{3}\cos(2A) + 1 = 4\sin\left(\frac{\pi}{6} + A\right)\sin\left(\frac{\pi}{3} - A\right)$.
(Ⅰ) Find the value of angle $A$;
(Ⅱ) If $a = \sqrt{2}$ and $b \geq a$, find the range of values for $\sqrt{2}b - c$.

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