
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 $f(x) = 2\sin x (\sin x + \cos x)$, where $x \in \mathbb{R}$.

(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.

(Ⅱ) If $f\left( \frac{a}{2} \right) = 1 + \frac{3\sqrt{2}}{5}$, and $\frac{3\pi}{4} < a < \frac{5\pi}{4}$, find the value of $\cos a$.

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