
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 the line $l_{1}:y=x+a$ intersects with the line $l_{2}:y=2(x+1)$ and the curve $C:y=x+\ln x$ at points $A$ and $B$ respectively, find the minimum distance between points $A$ and $B$.

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