
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 number of ordered quadruples  $(a, b, c, d)$  where each of  $a, b, c,$  and  $d$  are (not necessarily distinct) elements of  $\{1, 2, 3, 4, 5, 6, 7\}$  and  $3abc + 4abd + 5bcd$  is even. For example,  $(2, 2, 5, 1)$  and  $(3, 1, 4, 6)$  satisfy the conditions.

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