
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 curve $y = x^{3}+ x-2$, the tangent line ${l_{1}}$ at point $P_{0}$ is parallel to the line $4x-y-1=0$, and point $P_{0}$ is in the third quadrant.

(1) Find the coordinates of $P_{0}$;

(2) If line $l \perp {l_{1}}$, and $l$ also passes through the tangent point $P_{0}$, find the equation of line $l$.

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