
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
A sequence  $\{a_n\}_{n\geq0}$  obeys the recurrence  $a_n=1+a_{n-1}+\alpha a_{n-2}$  for all  $n\geq2$  and for some  $\alpha>0$ . Given that  $a_0=1$  and  $a_1=2$ , compute the value of  $\alpha$  for which  $$ \sum_{n=0}^{\infty}\frac{a_n}{2^n}=10 $$  

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