
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
A uniform tetrahedron has the numbers $1$, $2$, $3$, and $4$ written on its four faces, respectively. It is thrown twice, and the numbers facing down after each throw are $b$ and $c$.  
$(1)$ Let $z=(b-3)^2+(c-3)^2$, find the probability of $z=4$;  

$(2)$ If the equation $x^2-bx-c=0$ has at least one root $x \in \{1,2,3,4\}$, then the equation is called a "beautiful equation". Find the probability that the equation is a "beautiful equation".

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