Determining the critical value in a likelihood ratio test between two means of a Poisson random variable given critical region $\alpha$

Question

I have a homework problem I am working on (reworking actually, as practice for the exam, but just can't figure it out again).

I have taken a sample of size 6 of a Poisson Random Variable $$x_1, ..., x_6 $$ the null hypothesis is that $$X \sim Poi(0.5) $$

while the alternative is that $$X \sim Poi(1) $$

We used the Likelihood ratio test and Neyman Pearson lemma and decided that we can also find $c'$ of $$T = \Sigma_{k=1}^{6}x_k > c'$$ for $$R(x_1, ... , x_6) = \frac{H_a}{H_0}$$

so far so good, but for critical region $\alpha = 0.025 \leq P(T > c')$ we solve for c and get

$$c = 7$$

Now I have been at this all morning

I come to $$P(T > c'|H_0) = \Sigma_{k=c+1}^{\infty}e^{-3}3^k\frac{1}{k!}$$

I just can't find a way to isolate c. I see the Taylor expansion of e on the right minus the sum until c. But using that just ends up bringing me to the same dead-end from the other way around.

Can you help?

Thank you very much in advance.