
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  $ \left( x_n\right)_{n\ge 1} $  be a sequence having  $ x_1=3 $  and defined as  $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $  for every natural number  $ n. $  Find all values  $ m $  for which the terms  $ x_m,x_{m+1},x_{m+2} $  are in arithmetic progression, where  $ \lfloor\rfloor $  denotes the integer part.

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