
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 a sequence  $(a_n)$  by  $a_0 =0$  and  $a_n = 1 +\sin(a_{n-1}-1)$  for  $n\geq 1$ . Evaluate  $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k. $$  

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