
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
Consider a sequence of number pairs $P$: $(a_1,b_1), (a_2,b_2), \ldots, (a_n,b_n)$. Define $T_1(P) = a_1 + b_1$ and $T_k(P) = b_k + \max\{T_{k-1}(P), a_1 + a_2 + \ldots + a_k\}$ for $2 \leq k \leq n$, where $\max\{T_{k-1}(P), a_1 + a_2 + \ldots + a_k\}$ denotes the maximum of $T_{k-1}(P)$ and the sum $a_1 + a_2 + \ldots + a_k$.
- **(I)** For the sequence of number pairs $P$: $(2,5), (4,1)$, find the values of $T_1(P)$ and $T_2(P)$.
- **(II)** Let $m$ be the minimum of the numbers $a, b, c, d$. For a sequence of number pairs $P$: $(a, b), (c, d)$ and $P'$: $(c, d), (a, b)$, compare $T_2(P)$ and $T_2(P')$ for the cases $m=a$ and $m=d$.
- **(III)** Among all possible sequences of the five number pairs $(11,8), (5,2), (16,11), (11,11), (4,6)$, write down a sequence $P$ that minimizes $T_5(P)$ and provide the value of $T_5(P)$ (only a conclusion is needed).

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