
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
Acute triangle  $ABC$  has altitudes  $AD$ ,  $BE$ , and  $CF$ .  Point  $D$  is projected onto  $AB$  and  $AC$  to points  $D_c$  and  $D_b$  respectively.  Likewise,  $E$  is projected to  $E_a$  on  $BC$  and  $E_c$  on  $AB$ , and  $F$  is projected to  $F_a$  on  $BC$  and  $F_b$  on  $AC$ .  Lines  $D_bD_c$ ,  $E_cE_a$ ,  $F_aF_b$  bound a triangle of area  $T_1$ , and lines  $E_cF_b$ ,  $D_bE_a$ ,  $F_aD_c$  bound a triangle of area  $T_2$ .  What is the smallest possible value of the ratio  $T_2/T_1$ ?

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