
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
Suppose \( P(x) = a_{0} + a_{1} x + a_{2} x^{2} + \ldots + a_{8} x^{8} \) is a polynomial of degree 8 with real coefficients \( a_{0}, a_{1}, \ldots, a_{8} \). If \( P(k) = \frac{1}{k} \) when \( k = 1, 2, \ldots, 9 \), and \( b = P(10) \), find the value of \( b \).

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