
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 $(xOy)$, an ellipse ${{C}_{1}} :\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > b > 0 )$ has its left focus at ${{F}_{1}}\left( -1,0 \right)$ and the point $P\left( 0,1 \right)$ 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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