
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 math contest consists of  $9$  objective type questions and  $6$  fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly  $14$  out of  $15$  questions. Let  $O_1, O_2, \dots , O_9$  be the nine objective questions and  $F_1, F_2, \dots , F_6$  be the six fill inthe blanks questions. Let  $a_{ij}$  be the number of students who attemoted both questions  $O_i$  and  $F_j$ . If the sum of all the  $a_{ij}$  for  $i=1, 2,\dots , 9$  and  $j=1, 2,\dots , 6$  is  $972$ , then find the number of students who took the test in the school. 

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