
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
The line $L$ passes through point $P(1, 2)$, and the midpoint of the line segment $AB$ intercepted by two lines $L_1: 3x-y+2=0$ and $L_2: x-2y+1=0$ is exactly point $P$. Find the equation of line $L$.

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