
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 all arithmetical sequences of real numbers  $(x_i)^{\infty}=1$  and  $(y_i)^{\infty} =1$  with the common first term, such that for some  $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$ , and  $x_{k+1}y_{k+1} = 16$ . Find all such pairs of sequences with the maximum possible  $k$ .

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