
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 that the vertex of the parabola is at the origin, its axis of symmetry is the $x$-axis, and the distance from a point $P(3,a)$ on the parabola to the focus is $5$.

$(1)$ Find the standard equation of the parabola;  
$(2)$ Given that line $l$ passes through the fixed point $P(-3,1)$ with a slope of $k$, for what value(s) of $k$ does line $l$ have only one common point with the parabola, and write the equation of the corresponding line $l$.

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