
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
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, satisfying ${S}_{n}=2{a}_{n}-1(n∈{N}^{*})$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Solve the inequality in terms of $n$: ${a}_{1}{C}_{n}^{0}+{a}_{2}{C}_{n}^{1}+{a}_{3}{C}_{n}^{2}+{a}_{4}{C}_{n}^{3}+⋅⋅⋅+{a}_{n+1}{C}_{n}^{n}＜2023$.

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