
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
Assume a sequence $\{a_n\}$ satisfies: (i) $a_1=1$; (ii) all terms $a_n \in \mathbb{N}^*$; (iii) $1=a_1 < a_2 < \ldots < a_n < a_{n+1} < \ldots$. Define the set $A_m=\{n|a_n \leq m, m \in \mathbb{N}^*\}$, and denote the maximum element of set $A_m$ as $b_m$. In other words, $b_m$ is the maximum number of terms of the sequence $\{a_n\}$ that satisfy the inequality $a_n \leq m$. We call the sequence $\{b_n\}$ the accompanying sequence of $\{a_n\}$. For example, the accompanying sequence of $1, 3, 5$ is $1, 1, 2, 2, 3$.
(1) If the accompanying sequence of $\{a_n\}$ is $1, 1, 1, 2, 2, 2, 3$, provide the original sequence $\{a_n\}$.
(2) Let $a_n=3^{n-1}$, find the sum of the first $100$ terms of the accompanying sequence $\{b_n\}$.
(3) If the sum of the first $n$ terms of $\{a_n\}$ is $S_n= \frac{3}{2}n^2 - \frac{1}{2}n + c$ where $c$ is a constant, find the sum of the first $m$ terms of the accompanying sequence $\{b_n\}$, denoted as $T_m$.

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