
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 \( P_{i}(x) = x^2 + b_{i} x + c_{i} \) be real coefficient polynomials (\( b_{i}, c_{i} \in \mathbb{R}, i = 1, 2, \ldots, n \)) that are distinct from each other, and for any \( 1 \leq i < j \leq n \), the polynomial \( P_{i}(x) + P_{j}(x) \) has exactly one real root. Find the maximal value of \( n \).

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