
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
Two signals, $A$ and $B$, are transmitted over a communication line with probabilities of 0.72 and 0.28, respectively. Due to noise, $\frac{1}{6}$ of $A$ signals are distorted and received as $B$ signals, and $\frac{1}{7}$ of transmitted $B$ signals are received as $A$ signals.

a) Determine the probability that an $A$ signal will be received at the receiving end.

b) Given that an $A$ signal is received, what is the probability that it was originally transmitted as an $A$ signal?

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