
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
There are $n$ numbers, the first one is denoted as $a_{1}$, the second as $a_{2}$, ..., the $n$-th as $a_{n}$. If $a_{1}= \frac {1}{2}$, and starting from the second number, each number equals the reciprocal of the difference between $1$ and the number before it.  
$(1)$ Calculate the values of $a_{2}$, $a_{3}$, and $a_{4}$;  
$(2)$ Based on the results of $(1)$, conjecture and write down the values of $a_{2010}$, $a_{2011}$, and $a_{2012}$.

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