
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 ray of light starts from point $F_1(-1,0)$, reflects off a point $M$ on the line $l: x+2y+6=0$, and then passes exactly through point $F_2(1,0)$.
(1) Find the coordinates of the symmetric point $F'_1$ of point $F_1$ with respect to the line $l$;
(2) Find the equation of the ellipse $C$ that passes through point $M$ and has $F_1$ and $F_2$ as its foci.

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