
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 a sequence $\{a_n\}$ that satisfies $n \geqslant 2$, $a^{2}_{n-1}+2a_{n}=a_{n}^{2}+1$, and $a_{1}=2$, $a_{n} > 1$  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Calculate the value of $T_{n}=a_{1} \cdot 2^{a_{1}}+a_{2} \cdot 2^{a_{2}}+\ldots+a_{n} \cdot 2^{a_{n}}$.

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