
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 $\overrightarrow{m} = (1,1)$, vector $\overrightarrow{n}$ makes an angle of $\frac{3\pi}{4}$ with vector $\overrightarrow{m}$, and $\overrightarrow{m} \cdot \overrightarrow{n} = -1$.  
(1) Find the vector $\overrightarrow{n}$.  
(2) If the angle between vector $\overrightarrow{n}$ and vector $\overrightarrow{q} = (1,0)$ is $\frac{\pi}{2}$, and vector $p = (\cos x, 2\cos^2(\frac{\pi}{3} - \frac{x}{2}))$, where $0 < x < \frac{2\pi}{3}$, find the range of values for $|\overrightarrow{n} + \overrightarrow{p}|$.

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