
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
Find the maximum possible value of  $k$  for which there exist distinct reals  $x_1,x_2,\ldots ,x_k $  greater than  $1$  such that for all  $1 \leq i, j \leq k$ ,  $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$  *Proposed by Morteza Saghafian*

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