The country of Terniquaternaria follows an unusual numeral system. Its numbers are expressed as in base 4, but there are only three “digits”: 0, 1 and T, which has value (-1). For instance, the number “1T01” corresponds to the number 49 in the usual decimal system (i.e. +1*4^3-1*4^2+0*4^1+1*4^0). Due to the lack of a 4th digit, many numbers, such as 29, have no direct representation. Instead, the inhabitants of Terniquaternaria (called “Trits”) express them as the quotient of two representable numbers (when possible). For instance, 29 could be written as (1100T/1TT) or (11111T/1T0T), among other variants.
Decimus just landed in Terniquaternaria and needs to write down his passport number (524293) on the admission form. Is it possible? If so, how can it be written? If not, can you prove impossibility?