
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
A point $(\frac{\pi}{4}, 1)$ lies on the graph of the function $f(x) = 2a\sin x\cos x + \cos 2x$.
(Ⅰ) Determine the value of $a$ and the smallest positive period of $f(x)$.
(Ⅱ) Find the intervals where $f(x)$ is monotonically decreasing on $(0, \pi)$.

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