
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
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $sinC = sinAcosB + \frac{{\sqrt{2}}}{2}sin(A+C)$, find:

$(1)$ the value of angle $A$;

$(2)$ based on the original conditions, if one of the following conditions is added, determine which one can uniquely determine $\triangle ABC$. If the triangle is uniquely determined, find the height $h$ on side $BC$.

Condition 1: $a=2$, $sinC=\frac{{\sqrt{3}}}{2}$;

Condition 2: $a=\sqrt{5}$, $b=\sqrt{2}$.

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