
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
The coordinates of vertex \( C(x, y) \) of triangle \( \triangle ABC \) satisfy the inequalities \( x^{2}+y^{2} \leq 8+2y \) and \( y \geq 3 \). The side \( AB \) is on the x-axis. Given that the distances from point \( Q(0,1) \) to the lines \( AC \) and \( BC \) are both 1, find the minimum area of \( \triangle ABC \).

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