
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
A function \( f \), defined on the set of integers, satisfies the following conditions:

1) \( f(1) + 1 > 0 \)
2) \( f(x + y) - x f(y) - y f(x) = f(x)f(y) - x - y + xy \) for any \( x, y \in \mathbb{Z} \)
3) \( 2f(x) = f(x + 1) - x + 1 \) for any \( x \in \mathbb{Z} \)

Find \( f(10) \).

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