
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 two sequences \( a_{1} > a_{2} > \ldots > a_{n} \) and \( b_{1} < b_{2} < \ldots > b_{n} \) together contain each of the numbers \( 1, 2, \ldots, 2n \) exactly once. Determine the value of the sum

\[ \left|a_{1} - b_{1}\right| + \left|a_{2} - b_{2}\right| + \ldots + \left|a_{n} - b_{n}\right| \]

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