
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
For each  $k$ ,  $\mathcal{C}_k$  is biased so that, when tossed, it has probability  $\tfrac{1}{(2k+1)}$  of falling heads. If the  $n$  coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function  $n$ .

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