
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) Find the numbers  $a_0,. . . , a_{100}$ , such that  $a_0 = 0, a_{100} = 1$  and for all  $k = 1,. . . , 99$  :    $$ a_k = \frac12 a_{k- 1} + \frac12 a_{k+1 } $$  (b) Find the numbers  $a_0,. . . , a_{100}$ , such that   $a_0 = 0, a_{100} = 1$  and for all  $k = 1,. . . , 99$  :    $$ a_k = 1+\frac12 a_{k- 1} + \frac12 a_{k+1 } $$  .

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