
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 point $P(1,4)$, a line $L$ passes through it. The line intersects the positive semi-axes of $x$ and $y$ at points $A$ and $B$ respectively, with $O$ as the origin.

1. Denote the area of $\triangle ABO$ as $S$. Find the minimum value of $S$ and the equation of line $L$ in this case.
2. Find the equation of line $L$ when the sum of the distances $|OA|$ and $|OB|$ is minimized.

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