
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
Bucket  $A$  contains 3 litres of syrup. Bucket  $B$  contains  $n$  litres of water. Bucket  $C$  is empty.
We can perform any combination of the following operations:
- Pour away the entire amount in bucket  $X$ ,
- Pour the entire amount in bucket  $X$  into bucket  $Y$ ,
- Pour from bucket  $X$  into bucket  $Y$  until buckets  $Y$  and  $Z$  contain the same amount.**(a)** How can we obtain 10 litres of 30% syrup if  $n = 20$ ?**(b)** Determine all possible values of  $n$  for which the task in (a) is possible.

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