
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 sequences of positive integers \(1,\alpha_2, \alpha_3, \ldots\) and \(1,\beta_2, \beta_3, \ldots\) are an increasing arithmetic sequence with common difference \(d\) and an increasing geometric sequence with common ratio \(r\), respectively. For these sequences, let \(\gamma_n=\alpha_n+\beta_n\).
It is given that there is an integer \(k\) such that \(\gamma_{k-1}=200\) and \(\gamma_{k+1}=2000\). Find \(\gamma_k\).

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