
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 sequences $\{a_n\}$ and $\{b_n\}$ with their respective sums of the first $n$ terms denoted as $S_n$ and $T_n$, it is known that $b_n - a_n = 2^n + 1$ and $S_n + T_n = 2^{n+1} + n^2 - 2$. Find the expression for $T_n$.

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