
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
Define \( L(x) = x - \frac{x^2}{2} \) for every real number \( x \).  If \( n \) is a positive integer, define \( a_n \) by 
\[ 
a_n = L \left( L \left( L \left( \cdots L \left( \frac{19}{n} \right) \cdots \right) \right) \right),
\]
where there are \( n \) iterations of \( L \). As \( n \) approaches infinity, what value does \( n \cdot a_n \) approach?

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