
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 $f(x,y)={({1+\frac{m}{y}})^x}(m＞0,y＞0)$.  
$(1)$ When $m=3$, find the term with the largest binomial coefficient in the expansion of $f\left(6,y\right)$;  
$(2)$ If $f(4,y)={a_0}+\frac{a_1}{y}+\frac{a_2}{y^2}+\frac{a_3}{y^3}+\frac{a_4}{y^4}$ and $a_{3}=32$, find $\sum_{i=0}^4{a_i}$.

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