
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)* Call a four-digit number  $(xyzt)_B$  in the number system with base  $B$  stable if  $(xyzt)_B = (dcba)_B - (abcd)_B$ , where  $a \leq b \leq c \leq d$  are the digits of  $(xyzt)_B$  in ascending order. Determine all stable numbers in the number system with base  $B.$ *b)* With assumptions as in *a*, determine the number of bases  $B \leq 1985$  such that there exists a stable number with base  $B.$ 

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