
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 $\{a_{n}\}$ where $a_{1}=14$ and $a_{n+1}=3a_{n}-4$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ Let ${b_n}=\frac{{{{(-1)}^n}{a_n}}}{{({{3^n}+1})({{3^{n+1}}+1})}}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. If there exists $n\in N^{*}$ such that $m\geqslant T_{n}$, find the range of values for $m$.

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