
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 Lucas numbers are defined by \( L_{0}=2 \), \( L_{1}=1 \), and \( L_{n+2}=L_{n+1}+L_{n} \) for every \( n \geq 0 \). There are \( N \) integers \( 1 \leq n \leq 2016 \) such that \( L_{n} \) contains the digit 1. Estimate \( N \). An estimate of \( E \) earns \( \lfloor 20-2|N-E|\rfloor \) or 0 points, whichever is greater.

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