
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  $a_{i}$  and  $b_{i}$ , where  $i \in \{1,2, \dots, 2005 \}$ , be real numbers such that the inequality  $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$  holds for all  $x \in \mathbb{R}$  and all  $i \in \{1,2, \dots, 2005 \}$ . Find the maximum possible number of positive numbers amongst  $a_{i}$  and  $b_{i}$ ,  $i \in \{1,2, \dots, 2005 \}$ .

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