
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) A person simultaneously bought two boxes of matches and put them in his pocket. Each time they needed to light a match, they randomly took one or the other box. After a while, upon finding that one of the boxes was empty, the person discovered that it was completely empty. What is the probability that there are still $k$ matches left in the other box, if the number of matches in the unopened box was $n$?

b) Using the result from part a), find the sum

$$
C_{2 n}^{n}+2 C_{2 n-1}^{n}+4 C_{2 n-2}^{n}+\ldots+2^{n} C_{n}^{n}
$$

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