
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
The sequence $\{a_{n}\}$ satisfies $\frac{1}{{3{a_1}}}+\frac{1}{{5{a_2}}}+\frac{1}{{7{a_3}}}+⋯+\frac{1}{{(2n+1){a_n}}}=\frac{n}{{2n+1}}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let $b_{n}=\frac{{{a_n}+1}}{{{2^{{a_n}}}}}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}$. If for any positive integer $n$, the inequality $(3n+4)m≥(2n-5)({\frac{{16}}{9}-{S_n}})⋅{2^n}$ always holds, find the minimum value of $m$.

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