
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
Points  $H$  and  $T$  are marked respectively on the sides  $BC$  abd  $AC$  of triangle  $ABC$  so that  $AH$  is the altitude and  $BT$  is the bisectrix  $ABC$ . It is known that the gravity center of  $ABC$  lies on the line  $HT$ .
a) Find  $AC$  if  $BC$ =a nad  $AB$ =c.
b) Determine all possible values of  $\frac{c}{b}$  for all triangles  $ABC$  satisfying the given condition.

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