
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,x_2,\cdots)$  be a sequence of positive numbers such that  $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$  and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]

Determine real number  $ a$  such that if  $ x_1 > a$ , then the sequence is monotonically decreasing, and if  $ 0 < x_1 < a$ , then the sequence is not monotonic.

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