
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 vector $\overrightarrow{m} = (1, 1)$, the vector $\overrightarrow{n}$ forms an angle of $\frac{3\pi}{4}$ with the vector $\overrightarrow{m}$, and $\overrightarrow{m} \cdot \overrightarrow{n} = -1$. Also, the vector $\overrightarrow{n}$ is collinear with the vector $\overrightarrow{q} = (1, 0)$.
(Ⅰ) Determine the coordinates of vector $\overrightarrow{n}$
(Ⅱ) If the vector $\overrightarrow{p} = (2\cos^2 \frac{C}{2}, \cos A)$, where $A$ and $C$ are the internal angles of triangle $\triangle ABC$ with $\angle B = \frac{\pi}{3}$, find the range of values of $|\overrightarrow{n} + \overrightarrow{p}|$.

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