
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
In the infinite sequence \(\left\{x_n\right\}(n \geqslant 1)\), for every odd \(n\), \(x_n\), \(x_{n+1}\), and \(x_{n+2}\) form a geometric progression, while for every even \(n\), \(x_n\), \(x_{n+1}\), and \(x_{n+2}\) form an arithmetic progression. Given that \(x_1 = a\) and \(x_2 = b\):

1. Find the general term formula of the sequence. What necessary and sufficient conditions on real numbers \(a\) and \(b\) ensure the existence of such an infinite sequence?
2. Find the harmonic mean of \(x_2, x_4, \cdots, x_{2n}\), i.e., \(\frac{n}{\sum_{k=1}^{n} \frac{1}{x_{2k}}}\).

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