
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 the proposition $P(k)$: $\frac{1+a^2+a^4+\cdots+a^{2k}}{a+a^3+\cdots+a^{2k-1}} > \frac{k+1}{k}\left(a > 0, a\neq 1, k \in \mathbb{N}^*\right)$, determine the proposition $P(k+1)$.

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