
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
Find the total number of solutions to the following system of equations:

 $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} 
b(a + d)\equiv b \pmod{37} 
c(a + d)\equiv c \pmod{37} 
bc + d^2\equiv d \pmod{37} 
ad - bc\equiv 1 \pmod{37} \end{array}$ 

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