
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
Let  $x_1 \dots, x_{42}$ , be real numbers such that  $5x_{i+1}-x_i-3x_ix_{i+1}=1$  for each  $1 \le i \le 42$ , with  $x_1=x_{43}$ . Find all the product of all possible values for  $x_1 + x_2 + \dots + x_{42}$ .

*Proposed by Michael Ma*

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