
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 the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b>0$), points P$(x,y)$ and Q$(x',y')$ are two different points on the ellipse, except for the vertices. We have the following four inequalities:
1. $a^2+b^2\geq(x+y)^2$;
2. $\frac{1}{x^2}+\frac{1}{y^2}\geq\left(\frac{1}{a}+\frac{1}{b}\right)^2$;
3. $4\left(\frac{x}{a}\right)^2\leq\left(\frac{b}{y}\right)^2$;
4. $\frac{xx'}{a^2}+\frac{yy'}{b^2}\leq1$.

Which of the above inequalities always hold? (Fill in all correct proposition numbers).

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