
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  $f$  be a function defined on  $\{(i,j): i,j \in \mathbb{N}\}$  satisfying

(i)  $f(i,i+1)=\frac{1}{3}$  for all  $i$ (ii)  $f(i,j)=f(i,k)+f(k,j)-2f(i,k)f(k,j)$  for all  $k$  such that  $i <k<j$ .

Find the value of  $f(1,100)$ .

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