
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
In an arithmetic sequence $\{a_n\}$, $a_2=4$, and $a_4+a_7=15$.
$(1)$ Find the general term formula for the sequence $\{a_n\}$;
$(2)$ Let $b_n=2^{a_n-2}$, calculate the value of $b_1+b_2+b_3+\ldots+b_{10}$.

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