
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 a positive integer  $n$  and any real number  $c$ , define  $x_k$  recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix  $n$  and then take  $c$  to be the largest value for which  $x_{n+1}=0$ . Find  $x_k$  in terms of  $n$  and  $k,\; 1\le k\le n$ .

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