
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 ellipse $M$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, points $F_1(-1,0)$ and $C(-2,0)$ are respectively the left focus and the left vertex of the ellipse $M$. A line passing through point $F_1$ (not coinciding with the x-axis) intersects $M$ at points $A$ and $B$.  
(I) Find the standard equation of the ellipse $M$;  
(II) Determine whether there exists a line $l$ such that point $B$ lies on the circle with diameter $F_1C$. If it exists, find the equation of line $l$; if not, explain why.

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