
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
Read the following material: Xiao Ming, in order to calculate the value of $1+2+2^{2}+\ldots +2^{2017}+2^{2018}$, used the following method:<br/>Let $S=1+2+2^{2}+\ldots +2^{2017}+2^{2018}$①<br/>then $2S=2+2^{2}+\ldots +2^{2018}+2^{2019}$②<br/>②$-$① gives $2S-S=S=2^{2019}-1$<br/>$\therefore S=1+2+2^{2}+\ldots +2^{2017}+2^{2018}=2^{2019}-1$<br/>Please solve the following problems following Xiao Ming's method:<br/>$(1)1+2+2^{2}+\ldots +2^{9}=$______；<br/>$(2)3+3^{2}+\ldots +3^{10}=$______；<br/>$(3)$ Find the sum of $1+a+a^{2}+\ldots +a^{n}$ (where $a \gt 0$, $n$ is a positive integer, please provide the calculation process).

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