
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 points $A(1,2)$, $B(3,4)$, and lines $l_1: x=0$, $l_2: y=0$, and $l_3: x+3y-1=0$. Let $P_i$ be the point on line $l_i$ ($i=1,2,3$) such that the sum of the squared distances from $P_i$ to points $A$ and $B$ is minimized. Find the area of triangle $P_1P_2P_3$.

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