Mixed Effects Poisson Regression with Nesting?

Question

I have collected data for a study that had the following design:

Subjects walked through a food pantry five different days during one of two conditions: signs-down (no signs displayed, 2 days) and signs-up (signs displayed, 3 days) -- between-subjects variable.

Within-subjects: Within the signs-up condition, we had two types of signs displayed -- guided-play signs (with an active learning goal) or free-play signs (with no active learning goal).

Within-subjects: Within the guided-play condition, we had two signs -- signs that asked math questions and signs that asked questions about colors and shapes.

Within-subjects: Within the free-play condition, we had two signs -- signs that asked questions with only one-word answers and pronouncement signs (generic statements).

My DV here is a count variable of the number of certain kinds of behaviors families engage in during the observation period (qual_talk).

This is what my data looks like:

   subject_number type_signs_guided_free signs            qual_talk
            <dbl> <fct>                  <fct>                <dbl>
 1             51 guided play            math                     1
 2             51 guided play            colors & shapes          0
 3             51 free play              one word answers         1
 4             52 guided play            math                     1
 5             52 guided play            colors & shapes          0
 6             52 free play              pronouncements           0
 7             52 free play              one word answers         0
 8             53 guided play            math                     2
 9             54 guided play            colors & shapes          0
10             55 free play              pronouncements           0
11             56 guided play            math                    NA
12             56 free play              pronouncements           0
13             56 free play              one word answers         2
14             57 guided play            math                     1
15             57 guided play            colors & shapes          2
16             57 free play              pronouncements           0
17             57 free play              one word answers         0
18             58 guided play            colors & shapes          0
19             58 free play              pronouncements           0
20             58 free play              one word answers         0
# … with 326 more rows

Basically, I am interested in two within-subjects questions: (1) the effect of guided-play signs vs. free-play signs on the DV. (2) the effect of math signs vs. colors & shapes signs on the DV and one-word answers signs vs. pronouncements signs on the DV.

This is the current model I have, but I can't quite seem to capture the nesting (or multilevel) element just right. Here, I am trying to fit a mixed-effects Poisson regression model with qual_talk as the DV, type_signs_guided_free (guided vs. free play), child's gender, child's age as the fixed effects, and random intercepts by subject.

glmer(qual_talk ~ type_signs_guided_free + child1gender + target_child_age + (1|subject_number), family = poisson(link = "log"), data = foodpantry)

Any ideas on how to do this using glmer? If any part of my question is unclear, I would be happy to explain in more detail. The closest I got to an answer is this link, but it doesn't fully translate to my case.

Thanks in advance!