
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 all positive integer  $n$ , such that   $$ \left[\frac{n}{2^0}\right]\left[\frac{n}{2^1}\right]\ldots\left[\frac{n}{2^k}\right]+2\cdot 4^{[\frac{k}{2}]} $$  is a square, where  $k$  is the non-negative integer satisfying  $2^k\leq n<2^{k+1}$ . 

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