
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 $xOy$, given an ellipse $C_1$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ whose left focus is $F_1(-1,0)$, and the point $P(0,1)$ is on $C_1$.

$(1)$ Find the equation of the ellipse $C_1$.

$(2)$ Suppose a 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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