
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 a line $l$ passes through point $P(2,3)$, and is intersected by two parallel lines $l_{1}: 3x+4y-7=0$ and $l_{2}: 3x+4y+8=0$ to form a line segment of length $d$.
$(1)$ Find the minimum value of $d$;
$(2)$ Calculate the value of $d$ when line $l$ is parallel to the x-axis.

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