
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  $\mathcal{P}_1$  and  $\mathcal{P}_2$  be two parabolas with distinct directrices  $\ell_1$  and  $\ell_2$  and distinct foci  $F_1$  and  $F_2$  respectively. It is known that  $F_1F_2||\ell_1||\ell_2$ ,  $F_1$  lies on  $\mathcal{P}_2$ , and  $F_2$  lies on  $\mathcal{P}_1$ . The two parabolas intersect at distinct points  $A$  and  $B$ . Given that  $F_1F_2=1$ , the value of  $AB^2$  can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

[i]Proposed by Yannick Yao

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