
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
Determine whether the following statements are true. If they are true, provide proofs.

1. If \(P\) and \(Q\) are two different points on the same side of line \(l\), then there must exist two distinct circles that pass through points \(P\) and \(Q\) and are tangent to line \(l\).
2. If \(a > 0\), \(b > 0\), \(a \neq 1\), and \(b \neq 1\), then \(\log_a b + \log_b a \geqslant 2\).
3. Let \(A\) and \(B\) be two sets of points in the coordinate plane. Define \(C_r = \{(x, y) \mid x^2 + y^2 \leq r^2\}\). If for every \(r \geq 0\), \(C_r \cup A \subseteq C_r \cup B\), then \(A \subseteq B\).

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