
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 vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ that satisfy: $|\overrightarrow{x}| = 1$, $|\overrightarrow{y}| = 2$, and $(\overrightarrow{x} - 2\overrightarrow{y}) \cdot (2\overrightarrow{x} - \overrightarrow{y}) = 5$.

1. Find the angle $\theta$ between $\overrightarrow{x}$ and $\overrightarrow{y}$.
2. If $(\overrightarrow{x} - m\overrightarrow{y}) \perp \overrightarrow{y}$, find the value of the real number $m$.

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