
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 the sequence $\left\{a_{n}\right\}$, defined by:
$$
a_{1}=\frac{b(b+1)}{2}+c,
$$

where $b \in \mathbf{N}_{+}$, $c \in \mathbf{N}$, and $0 \leqslant c \leqslant b-1$, and
$$
a_{n+1}=\left\{\begin{array}{ll}
a_{n}+n, & a_{n} \leqslant n; \\
a_{n}-n, & a_{n}>n \quad (n=1,2, \cdots) .
\end{array} \right.
$$

The first fixed term is $T_{1}=b+2c$, and its general form is
$$
T_{m}=\frac{(2b+4c-1)3^{m-1}+1}{2} \quad (m \in \mathbf{N}_{+}).
$$

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