
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
$a_1 = \frac{1}{2}$,  
$a_2 = \frac{1}{3}(1-a_1) = \frac{1}{6}$,  
$a_3 = \frac{1}{4}(1-a_1-a_2) = \frac{1}{12}$,  
$a_4 = \frac{1}{5}(1-a_1-a_2-a_3) = \frac{1}{20}$,  
...  
Following this pattern, when $n \in \mathbb{N}^*$, $a_n =$ \_\_\_\_\_\_.

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