
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
Consider two sequences where the first sequence, $1, a_2, a_3, \dots$ is an arithmetic sequence, and the second sequence, $1, b_2, b_3, \dots$ is a geometric sequence. If $c_n = a_n + b_n$ denotes the sum of the $n$-th terms of these sequences, and there is an integer $k$ such that $c_{k-1}=150$ and $c_{k+1}=900$, determine the value of $c_k$.

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