
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
Let the curve of the function $y = x^2 + 1$ have two points $P_1$ and $P_2$, with their projections on the $x$-axis being $Q_1$ and $Q_2$, respectively. Determine these points such that the tangent line at $P_1$ passes through $Q_2$ and the tangent line at $P_2$ passes through $Q_1$.

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