
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 $x_1, x_2 \in \mathbb{R}$, determine whether the condition "$x_1 > 1$ and $x_2 > 1$" is necessary and sufficient for the conditions "$x_1 + x_2 > 2$ and $x_1x_2 > 1$".

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