
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 \equal{} 1$  and  $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$  for  $ n \geq 1$ . Let  $ k$  be the least integer greater than  $ 1$  for which  $ a_k$  is an integer. Find  $ k$ .

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