
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 that a geometric sequence $\{x_n\}$ consists of positive numbers not equal to 1, and the sequence $\{y_n\}$ satisfies $y_n\log_{x_n}a=2$ ($a>0$, $a\neq 1$). Let $y_3=18$, $y_6=12$.
(1) Up to which term does the sum of the sequence $\{y_n\}$ reach its maximum value and what is this maximum value?
(2) Determine whether there exists a natural number $M$ such that $x_n > 1$ holds for all $n > M$. If such an $M$ exists, find it; if not, provide an explanation.
(3) Define a sequence $\{a_n\} = \log_{x_n}x_{n+1}$ (for $n > 13$, $n\in \mathbb{N}$). Discuss the monotonicity of the sequence $\{a_n\}$.

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