
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 number $x$ is chosen at random from the set $\{1,2,3,\cdots,12\}$, and then $y$ is chosen at random from the same set. The probability that the integer $4^x+5^y$ has units digit $1$ is:
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{3}{4}$
D) $1$

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