
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
The function \( g \), defined on the set of integers, satisfies the following conditions:

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

Find \( g(5) \).

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