
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 sequence \(\{a_n\}\) satisfies \(a_1 = 1, a_2 = \frac{1}{2}\), and \(a_n(a_{n-1} + a_{n+1}) = 2 a_{n+1} a_{n-1}\) for \(n \geqslant 2\). Find the 2012th term of the sequence \(\{a_n\}\).

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