
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
Let  $P$  be a  $10$ -degree monic polynomial with roots  $r_1, r_2, . . . , r_{10} \ne $  and let  $Q$  be a  $45$ -degree monic polynomial with roots  $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$  where  $i < j$  and  $i, j \in \{1, ... , 10\}$ . If  $P(0) = Q(1) = 2$ , then  $\log_2 (|P(1)|)$  can be written as  $a/b$  for relatively prime integers  $a, b$ . Find  $a + b$ .

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