
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
Yan and Jacob play the following game. Yan shows Jacob a weighted 4-sided die labelled 1, 2, 3, 4, with weights $\frac{1}{2}, \frac{1}{3}, \frac{1}{7}, \frac{1}{42}$, respectively. Then, Jacob specifies 4 positive real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ such that $x_{1}+\cdots+x_{4}=1$. Finally, Yan rolls the dice, and Jacob earns $10+\log \left(x_{k}\right)$ dollars if the die shows $k$. Which $x_{i}$ should Jacob pick to maximize his expected payoff?

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