
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  $a_1, a_2, \dots, a_n$  be real numbers, and let  $b_1, b_2, \dots, b_n$  be distinct positive integers. Suppose that there is a polynomial  $f(x)$  satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for  $f(1)$  in terms of  $b_1, b_2, \dots, b_n$  and  $n$  (but independent of  $a_1, a_2, \dots, a_n$ ).

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