
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
For integer  $a$ ,  $a \neq 0$ ,  $v_2(a)$  is greatest nonnegative integer  $k$  such that  $2^k | a$ . For given  $n \in \mathbb{N}$  determine highest possible cardinality of subset  $A$  of set  $ \{1,2,3,...,2^n \} $  with following property:
For all  $x, y \in A$ ,  $x \neq y$ , number  $v_2(x-y)$  is even.

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