
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 the sequence $\{a_n\}$ is a geometric sequence, and $a_4 = e$, if $a_2$ and $a_7$ are the two real roots of the equation $$ex^2 + kx + 1 = 0, (k > 2\sqrt{e})$$ (where $e$ is the base of the natural logarithm),
1. Find the general formula for $\{a_n\}$.
2. Let $b_n = \ln a_n$, and $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. When $S_n = n$, find the value of $n$.
3. For the sequence $\{b_n\}$ in (2), let $c_n = b_nb_{n+1}b_{n+2}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{c_n\}$. Find the maximum value of $T_n$ and the corresponding value of $n$.

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