
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 $f\left(x\right)=\left(1+2x\right)^{n}$, where the sum of the binomial coefficients in the expansion is $64$, and ${\left(1+2x\right)^n}={a_0}+{a_1}x+{a_2}{x^2}+…+{a_n}{x^n}$.  
$(1)$ Find the value of $a_{2}$;  
$(2)$ Find the term with the largest binomial coefficient in the expansion of $\left(1+2x\right)^{n}$;  
$(3)$ Find the value of $a_{1}+2a_{2}+3a_{3}+\ldots +na_{n}$.

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