
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
On a blackboard the product  $log_{( )}[ ]\times\dots\times log_{( )}[ ]$  is written (there are 50 logarithms in the product). Donald has  $100$  cards:  $[2], [3],\dots, [51]$  and  $(52),\dots,(101)$ . He is replacing each  $()$   with some card of form  $(x)$  and each  $[]$  with some card of form  $[y]$ . Find the difference between largest and smallest values Donald can achieve.

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