
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
Three people, A, B, and C, are shooting at a target. Their respective hit rates are $p_{1}$, $p_{2}$, and $\frac{1}{3}$. If they all shoot at the same target, the probability that A and C hit the target while B does not hit the target is $\frac{1}{18}$, and the probability that B hits the target while C does not hit the target is $\frac{4}{9}$. Given that "A hits the target," "B hits the target," and "C hits the target" are independent events, determine the values of $p_{1}$ and $p_{2}$.

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