
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
Given that  $ x^{4} - 2^{ - y^{2} } x^{2} - \left\| x^{2} \right\| + 1 \equal{} 0$, where  $ \left\| a\right\|$  denotes the greatest integer not exceeding  $ a$, find the number of pairs of real numbers  $ \left(x,y\right)$.

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