
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
Suppose  $ABC$  is a triangle, and define  $B_1$  and  $C_1$  such that  $\triangle AB_1C$  and  $\triangle AC_1B$  are isosceles right triangles on the exterior of  $\triangle ABC$  with right angles at  $B_1$  and  $C_1$ , respectively. Let  $M$  be the midpoint of  $\overline{B_1C_1}$ ; if  $B_1C_1 = 12$ ,  $BM = 7$  and  $CM = 11$ , what is the area of  $\triangle ABC$ ?

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