
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
Let $\triangle ABC$ have the internal angles A, B, and C opposite to the sides of length a, b, and c respectively, and let $\mathbf{m} = (\cos A, \cos C)$, $\mathbf{n} = (c - 2b, a)$, and $\mathbf{m} \perp \mathbf{n}$.
(Ⅰ) Find the magnitude of angle A.
(Ⅱ) If angle B = $\frac{\pi}{3}$ and the median AM on side BC of $\triangle ABC$ has a length d, find the area of $\triangle ABC$.

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