
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 circle defined by the equation $x^2 + y^2 - 2x + 4y + 1 = 0$, and point $Q$ lies within the region defined by the system of inequalities $ \begin{cases} xy \geqslant 2 \\ x - y \geqslant 0 \\ y \leqslant 1 \end{cases}$, find the minimum length of the segment $PQ$.

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