
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
A school has $2500$ students, including $1000$ in grade 10, $900$ in grade 11, and $600$ in grade 12. To understand the students' physical health, a stratified sampling method is used. If $100$ students are selected from the school, and the sample sizes from grade 10 and grade 12 are $a$ and $b$ respectively, the line $ax + by + 8 = 0$ intersects with the circle centered at $A(1, -1)$ at points $B$ and $C$, and $\angle BAC = 120^\circ$. Find the equation of circle $C$.

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