
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
1. The negation of the statement "There exists an $x \in \mathbb{R}$ such that $x^{2}-3x+3=0$" is true.
2. The statement "-1/2 < x < 0" is a necessary but not sufficient condition for "$2x^{2}-5x-3 < 0$".
3. The negation of the statement "If $xy=0$, then at least one of $x$ or $y$ is equal to $0$" is true.
4. The curves $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ and $\frac{x^{2}}{25-k}+\frac{y^{2}}{9-k}=1 (9 < k < 25)$ share the same foci.
5. There exists a unique line that passes through the point $(1,3)$ and is tangent to the parabola $y^{2}=4x$.

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