
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
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x^2 < a^2$ and $y^2 < b^2$. How many of the following inequalities must be true?
$\textbf{(I)}\ x^2+y^2 < a^2+b^2\qquad$
$\textbf{(II)}\ x^2-y^2 < a^2-b^2\qquad$
$\textbf{(III)}\ x^2y^2 < a^2b^2\qquad$
$\textbf{(IV)}\ \frac{x^2}{y^2} < \frac{a^2}{b^2}$
**A)** 0   
**B)** 1   
**C)** 2   
**D)** 3   
**E)** 4

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