
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  $ABC$  be a scalene triangle. Let  $I_0=A$  and, for every positive integer  $t$ , let  $I_t$  be the incenter of triangle  $I_{t-1}BC$ . Suppose that the points  $I_0,I_1,I_2,\ldots$  all lie on some hyperbola  $\mathcal{H}$  whose asymptotes are lines  $\ell_1$  and  $\ell_2$ . Let the line through  $A$  perpendicular to line  $BC$  intersect  $\ell_1$  and  $\ell_2$  at points  $P$  and  $Q$  respectively. Suppose that  $AC^2=\frac{12}{7}AB^2+1$ . Then the smallest possible value of the area of quadrilateral  $BPCQ$  is  $\frac{j\sqrt{k}+l\sqrt{m}}{n}$  for positive integers  $j$ ,  $k$ ,  $l$ ,  $m$ , and  $n$  such that  $\gcd(j,l,n)=1$ , both  $k$  and  $m$  are squarefree, and  $j>l$ . Compute  $10000j+1000k+100l+10m+n$ .

*Proposed by Gopal Goel, Luke Robitaille, Ashwin Sah, & Eric Shen*

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