
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
Definition: A sequence $\{x_n\}$ is defined as a "p-oscillating sequence" if there exists a constant $p$ such that for any positive integer $n$, $(x_{n+1} - p)(x_n - p) < 0$ holds.

1. For the sequences $\{a_n\} = 2n - 1$ and $\{b_n\} = q^n$ (-1 < q < 0), $n \in \mathbb{N}^*$, determine whether they are "p-oscillating sequences" or not.
2. Given a "p-oscillating sequence" $\{c_n\}$ that satisfies $c_{n+1} = \frac{1}{c_n + 1}$ and $c_1 = 1$, find the value of the constant $p$.

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