
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
In the Cartesian coordinate system xOy, it is known that the ellipse $C_1$: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) has its left focus at $F_1(-1,0)$, and the point $P(0,1)$ is on $C_1$.  
(1) Find the equation of the ellipse $C_1$;  
(2) Suppose the line $l$ is tangent to both the ellipse $C_1$ and the parabola $C_2$: $y^{2}=4x$. Find the equation of the line $l$.

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