
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
Find the pattern.<br/>Group $1$: $\frac{1}{1}+\frac{1}{3}=\frac{4}{3}$, $4^{2}+3^{2}=5^{2}$;<br/>Group $2$: $\frac{1}{3}+\frac{1}{5}=\frac{8}{15}$, $8^{2}+15^{2}=17^{2}$;<br/>Group $3$: $\frac{1}{5}+\frac{1}{7}=\frac{12}{35}$, $12^{2}+35^{2}=37^{2}$;<br/>$\ldots \ldots $<br/>$(1)$ Please write down the equation for Group $4$______,______;<br/>$(2)$ Please write down the equation for Group $n$______,______;<br/>$(3)$ If $k^{2}+9603^{2}=9605^{2}(k \gt 0)$, then $k=\_\_\_\_\_\_$.

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