
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  $0<c<1$  be a given real number. Determine the least constant  $K$  such that the following holds: For all positive real  $M$  that is greater than  $1$ , there exists a strictly increasing sequence  $x_0, x_1, \ldots, x_n$  (of arbitrary length) such that  $x_0=1, x_n\geq M$  and
\[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\]

(From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

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