
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 nine non-negative real numbers \(a_{1}, a_{2}, \cdots, a_{9}\) whose sum is 1, define
$$
\begin{array}{l}
S=\min \left\{a_{1}, a_{2}\right\}+2 \min \left\{a_{2}, a_{3}\right\}+\cdots+8 \min \left\{a_{8}, a_{9}\right\}+9 \min \left\{a_{9}, a_{1}\right\}, \\
T=\max \left\{a_{1}, a_{2}\right\}+2 \max \left\{a_{2}, a_{3}\right\}+\cdots+8 \max \left\{a_{8}, a_{9}\right\}+9 \max \left\{a_{9}, a_{1}\right\},
\end{array}
$$

where \(\min \{x, y\}\) denotes the smaller of \(x\) and \(y\), and \(\max \{x, y\}\) denotes the larger of \(x\) and \(y\). Let \(S_{0}\) be the maximum possible value of \(S\). When \(S=S_{0}\), find all possible values of \(T\).

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