
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 $2, a_2, a_3, \dots$ and $1, b_2, b_3, \dots$ are an increasing arithmetic sequence and an exponential sequence, respectively. If for the sequence $c_n = a_n + b_n$, there is an integer $k$ such that $c_{k-1}=400$ and $c_{k+1}=1600$. Find $c_k$.

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