
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
 $a_0, a_1, \ldots, a_{100}$  and  $b_1, b_2,\ldots, b_{100}$  are sequences of real numbers, for which the property holds: for all  $n=0, 1, \ldots, 99$ , either  $$ a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n, $$  or  $$ a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n. $$  Given  $a_{100}\leq a_0$ , what is the maximal value of  $b_1+b_2+\cdots+b_{100}$ ?

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