
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: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]

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