When child care expenses are split 50/50

My ex and I split physical custody of our son 50/50.  We also split his daycare expenses 50/50.  For tax purposes, I am the custodial parent (he lives with me 183 days per year), so I will be claiming daycare expenses.  Since we split daycare expenses 50/50, can I only claim the portion of those expenses that I pay?  If so, it seems we're leaving half of the child care expenses "unclaimed". 

 

It seems like I should be able to claim 100% of the daycare expenses and then immediately reimburse my ex for half of the tax credit.  Otherwise, in cases where parents have agreed to split child care expenses 50/50, it would make more sense (tax-wise) for the custodial parent to pay 100% of those expenses and then have the non-custodial parent immediately reimburse him/her for half each month.  Right?