
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
Let  $F_n$  be the set of all bijective functions  $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$  that satisfy the conditions:

1.  $f(k)\leq k+1$  for all  $k\in\left\{1,2,\ldots,n\right\}$ 2.  $f(k)\neq k$  for all  $k\in\left\{2,3,\ldots,n\right\}$ Find the probability that  $f(1)\neq1$  for  $f$  randomly chosen from  $F_n$ .

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