
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 $f(x) = \sin x + a\cos x$, one of its zeros is $\frac{3\pi}{4}$.

(Ⅰ) Find the value of the real number $a$;  
(Ⅱ) Let $g(x) = [f(x)]^2 - 2\sin^2x$, find the intervals where $g(x)$ is monotonically increasing.

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